My pulse quickens as I drive on a backroad north of Amarillo. The dog, who has climbed into the passenger seat, looks up at me with big green eyes that someone once described as “extra-sentient.” He knows this isn’t good.

We aren’t supposed to be here, on this empty road. The day before, we had driven to Amarillo from Austin and stayed at the Big Texan Motel adjacent to the Big Texan Steak Ranch—kitsch lover’s paradise, low rates, only a $20 pet fee. The property features a “horse hotel” situated around the corner from our room. I think that’s why the dog kept me up for hours with his pacing; he was communing with the horses. If I’d gotten adequate rest, I’d like to think I’d have taken Interstate 40, as planned, to Albuquerque, New Mexico. There would have been ample opportunities to refuel.

I’m not driving my own car. It’s in the shop. A friend in Austin lent me his silver Beetle, whose fuel gauge, I am just learning, does not function properly. Twenty minutes ago, we had a third of a tank.

Leaving the motel this morning, I wasn’t paying attention to the map on my phone. I let Siri guide me. Now we’re out here, swallowed up by the plains with no towns in sight, the needle on the gauge falling, cell signal nonexistent. I tell the dog, wearily, this might be a long day.

Just how I acquired a dog from a hunter in South Dakota upon moving to Texas from New York five years earlier is a story too long and dense with characters. The short of it: A man handed him to me at a motel in Sioux Falls, and I flew with him in my arms back to Austin, to an empty rental full of boxes, where we started a new life together.

He has a white spot on the top of his head like a halo and a brighter orange patch over his left eye. His coat is apricot. The rest of him is speckled with white as though dusted with powdered sugar. I named him Laszlo. People often ask if it was after the Casablanca character Victor Laszlo (a hero). Less often people ask if it was after the artist László Moholy-Nagy (a genius). Neither. My father is Hungarian, and I always liked the name. Though mostly people call him Laz.

My favorite thing to do with this dog is to drive. Not city drives, but long ones where you start in one landscape and end up somewhere entirely different. For me, it is analogous to diving into the sea and swimming for a good while in open water before your return to shore, a conversation with one’s edges, a reminder of scale. In the glare of the sun as in the shadows of thunderclouds, when you’re far out on the road, life can feel like it is playing out in another dimension. It’s the anonymity and the isolation and the concurrent speed and stillness, and there’s something else I don’t yet know the word for. Whatever it is, I feel it most intensely while driving the highways and backroads of Texas, where the alchemy of so much out there and the frankness of strangers keep doors to the uncanny more ajar than in other places.

I suppose my penchant for long drives is inherited. One of my grandfathers was a geologist from Missouri whose work took him on highways alone across the Southwest. His favorite song was “King of the Road,” and his favorite road was US 84 through New Mexico. My other grandfather was a physician from Budapest, Hungary, who upon fleeing after a failed uprising found work in far Upstate New York. He was in awe of the vastness of his adopted country and, in 1963, took my grandmother and 6-year-old father on a road trip from near the Canadian border to Mexico City. He completed the whole drive in a suit and tie. Exquisite pictures from the destination survive, but I wonder about photos from the road, if they even took any. I wish I knew what my grandparents had made of Texas—they’d driven it north to south and back again. All my father remembers is getting sick and going to a hospital in Laredo.

Laszlo and I are still on the backroad as the needle on the fuel gauge enters the red zone. I am resigned to getting stuck on the side of the road. At least this place is visually compelling—the flat plains have given way to undulating ground with arroyos and red dirt that remind me of Arizona, where I’m from. But I must concentrate on not driving too fast so I can conserve gas. I see an intersection ahead—US 385—and feel optimistic. We turn, cross the Canadian River, and keep on north, hoping for fuel. Time slows.

Then, in the distance, a water tower comes into focus. I shout; Laz sits up. I turn off US 385. Channing: Pop. 356, says the sign. I see a gas station! No, it’s only the remnants of one, I realize and curse. I park the Beetle in front of a shuttered candle store named Just Great and step out into the hot air with Laszlo on a leash. I am relieved to find that my phone signal has reengaged. I dial AAA. I am on hold when a police car approaches. A woman from the sheriff’s office rolls down her window.

“Ran out of gas?” she asks. They apparently get a lot of stranded motorists out here. “I’ll call the county judge right now.” The county judge?

Maybe 15 minutes later, a red pickup truck arrives. A man with thick white hair climbs out: Hartley County Judge Ronnie Gordon. He asks if I have enough gas to follow him five blocks to the county diesel tank. Possibly not, but I’ll try.

We reach the tank. The judge tops me off so I can get to a gas station in Dalhart, 30 miles north. I am overwhelmed by our good luck but embarrassed not to have any cash on me. When you’re caught in a position of physical helplessness and the right stranger appears at the right time and bestows on you his generosity, that person becomes a folk hero in the theater of your memory. I mail him a $20 and a thank-you note. Time passes and I think of the encounter often.

Several months later, we move to Los Angeles. I go to a psychic. I eat an array of citrus. I complain about the traffic. Laz is never the same again. At first, I attribute it to homesickness.

An old man lives alone up the hill behind our apartment. At dusk most days he plays jazz on the loudspeakers in his garden. We sit outside and listen. One evening, our neighbor plays Willie Nelson’s Red Headed Stranger. I leave a note in his mailbox a couple days later telling him how much I enjoyed his music, and especially Willie because it reminded me of where I’d come from. The neighbor stops playing music after that.

I am homesick, too. Sometimes you don’t realize a place is home until you leave it. Call it a game of Twister with my own geography: Arizonan moves to Texas by way of New York only to feel, through living in California, drawn back to Texas.

After a hike one afternoon, Laszlo starts crying. His leg is swollen from hip to paw. He can’t walk, and I carry him to the car in my arms. Rattlesnake bite, the vet says. His fur never grows back at the fang marks. His behavior deteriorates. He barks and growls at people he knows and likes. The vet thinks it could be dementia. But he is only 6. I feel like I am starting to lose him.

I take Laz on a long drive out to the desert past Yucca Valley. Driving isn’t as enjoyable here. The speed limit is too low, the roads too crowded. It might take hours to get out of Los Angeles in the first place and, once you do, there are no county judges in red pickup trucks to save you if you run out of gas. That memory is a stand-in for everything I miss about Texas—the openness of its plains, the enormity of its sky, the quiet kindness of its strangers.

I decide to move back.

I am too late. A few weeks before we are set to return, Laszlo bites someone. My friend on a ranch in Colorado takes him in for good. He hosts a small band of dogs who have either started to lose it or who have lost their owners. Now they live carefree at the edge of the wilderness. I’ll visit, but still, I fear he is going to think I’ve died. I don’t want him to grieve me. As Joy Williams wrote in the essay “Hawk,” “Dogs are the best mourners in the world.”

As I settle back in Austin without him, Laszlo comes to me in dreams. We are driving across Texas: going west on Interstate 10 into the Franklin Mountains or speeding across the High Plains. It is always golden hour.

The sting of his absence doesn’t soften. I realize I need to do something. I consider calling the LA psychic but instead decide I need to drive, far.

I choose the Panhandle. I want to pay a visit to the county judge. Zooming in on the map as I plan my route, I can’t believe what I am seeing. One hundred miles south of Channing, where the judge saved me, are the letters Lazbuddie. Thinking this might be a new dream—Laszlo, Laz, Lazbuddie—I make my boyfriend consult the map. Lazbuddie is a real place, an unincorporated town in Parmer County. It is my sign.

Out of the prairie and into the plains on US 183, the palette and land flatten in tandem. I’m listening to the new translation of The Odyssey. I’m an enthusiast for mismatched sonic experiences. (Whoever else has gazed up at the Acropolis while listening to George Strait’s “Amarillo by Morning,” call me.) On State Highway 153 after Winters, Odysseus outsmarts Polyphemus just as I pass a wreck. I pull over. Two teenagers and a puppy have emerged from the flipped SUV unscathed. The older boy, who had fallen asleep at the wheel, paces the side of the highway and shouts into his phone, “We are in the middle of nowhere!” A couple in an RV waits with them for help. I call it a day three hours later, in Lubbock.

In the morning, I keep north on I-27 toward Amarillo. The best place-names are here: Happy, New Deal, Shallowater. I’ve never known a more accurately named town than Plainview.

The county judge told me over email that he’d be free at 2 p.m. at the Hartley County Courthouse. I have some time, so I stop by Cadillac Ranch. Parents are taking photographs of their children spray-painting the cars. At the entrance, a kid named Xavier is hawking tchotchkes in front of a van that has seen better days. I buy a cerulean magnet in the shape of Texas and keep it in my cup holder like an amulet.

I retrace my route from the day Laz and I ran out of gas. Making this road trip without him feels peculiar, like I left something important behind. At the Steak Ranch, a tall man holding a selfie stick and sporting a yellow T-shirt that bears the message “Written and Directed by Tarantino” in a Western font is filming himself outside the entrance. I pass the horse hotel and stop for gas. On the backroad past Amarillo, the land is as open and quiet as I remember. I can’t believe it when I see a stand-alone gas pump near Tascosa and a sign announcing 24 Hour Fuel. This has to be new. If it isn’t, then my memory of the day is cracked, and I’d taken US 87 and gone west on SH 354 to Channing.

About 15 minutes later, I see the water tower. I find it surprising for Channing to have no gas station but a courthouse. Built in 1906, it’s an exquisite one at that: stately brick and sandstone, clad with ionic columns, and surrounded by mowed grass and tall elms. I walk up creaky wooden stairs, pausing before an oil painting of a cowboy on horseback in the snow under an inky sky, the full moon lighting his way.

“Come on in,” I hear. It’s him. The county judge, dressed in a black-and-white gingham button-down, jeans, and boots, sits in a small office off a simple courtroom. Gordon speaks dryly and openly like the plains on which he has lived his whole life. “A lot of cowboys around here,” he says, “and cowboys are always willing to help you.”

Gordon has served as county judge for over 30 years. The old gas station, he tells me, shuttered about 20 years ago. Since then, he’s helped too many people to count. A few encounters stand out, like a motorcyclist stranded at night in a blizzard. Some preachers. And a rodeo star, which makes me feel better, somehow, knowing that a famous barrel racer would forget to refuel in Amarillo.

Gordon says this year has been slow by his measure. It is late May and he’s only helped five or six drivers so far. To me that sounds like plenty. I thank him again for helping us and tell him how the encounter came to symbolize one of the things I most appreciate about Texas: the unfussy generosity of people you meet along the way here. He doesn’t think much of it—“just a little bit of gas.” But, I say, where else does a county judge come to your rescue? “Yeah, yeah, I hear that a lot,” he admits. “And in a red pickup truck!” I wish him the good fortune of aiding noteworthy stranded motorists his final term as county judge (he’s decided not to run again). He is hoping for a NASCAR driver.

It is time to drive to Lazbuddie. I take US 385 south to Vega and keep going. This is dairy and meatpacking country, and the air is heavy. After Hereford (“Welcome to the Beef Capital of the World”), I turn onto US 60 and later onto a mostly empty backroad. Just me and a few bull haulers. Wheat fields spread out in all directions like a golden sea.

There is the sign: Lazbuddie 23.

Twenty minutes later, I arrive. I stand in front of the post office at the end of town, which is less than a mile from its start, and take pictures. A woman drives up to drop off some mail. As this isn’t a place people just stumble upon, I feel the need to tell her I’m here because my dog is Laz (I can’t bear the past tense). “That’s exactly how we say it!” she says. “We call Nazareth”—a nearby town—“Naz and Lazbuddie Laz.”

I want to drop something in the mail slot to get a Lazbuddie postmaster stamp. But I don’t have any postage and the post office is closed for the day. Across the street I notice a farm store that looks like it might also be a general store.

A man and woman are sitting at a big round table, and a younger man and woman stand behind the register—two generations of one family. This is their store. Its interiors are like time travel. They are amused by the idea of a dog named Laz. The seated man tells me he is 90 and has lived in Lazbuddie his entire life. The unincorporated town has no mayor. He is its unofficial mayor. The woman beside him, his wife, tells me his name is Ted, “but everyone calls him Mr. Laz.”

Mr. Laz tells me the story of Lazbuddie. This land had been part of the legendary XIT Ranch that was sold off to pay for the construction of the Texas State Capitol. Midwesterners, including his family, had moved out here upon seeing advertisements for parcels of land. The newcomers needed a post office, and for that, the place would need a name. A Luther who went by “Laz” and an Andrew who went by “Buddie” had opened a store called the Lazbuddie Commissary, and the name stuck.

Mr. Laz, bespectacled and wearing overalls over a baby blue polo shirt, stands up and shows me around the store, a maze of rooms. In his office, life-size cutouts of John Wayne are propped on one wall and a cutout of Elvis on another. I notice some guitars. “I taught my brother how to play,” he tells me. The family name is Treider but his brother, Larry, dropped the first “e” when he set his sights on becoming a musician. He knew Buddy Holly and was inspired to start out in rock ’n’ roll, but over time his sound became country. A few years ago, before his death, he was inducted into the West Texas Walk of Fame. His obituary says he had talent on par with Waylon Jennings, Willie Nelson, and Ray Price but just hadn’t been in the right place at the right time.

“I gave him a hard time about it,” Mr. Laz tells me of his brother’s turn to country music. “Sing a country song backward and you’ll get your car back, your wife back.” He starts singing “Tennessee Waltz.” “Know what I mean?”

Mr. Laz tells me the fastest way to Lubbock. Outside, he notices my vehicle registration sticker is expiring next month and reminds me not to forget. Looking around, he asks where Laz is. I tell him the story. “Oh well, I’m losing it, too,” he says. “Just haven’t bitten anyone yet. Give me a couple more years and they’ll take me out to pasture.”

Mr. Laz stands at the threshold of his store, waving slowly with slightly cupped fingers until my car fades from sight. This is the other dimension I was talking about.