Pierce each potato five or six times with the tines of a fork. Lightly coat the potatoes with vegetable oil and season liberally with salt and pepper. Lightly coat the tomatillos with vegetable oil, season with salt, and place on a small baking sheet to catch any dripping juices during cooking.

Follow the instructions on How to Light Your Pit, aiming for a temperature of 285°F. When the coal is ready, add two or three chunks of pecan and let the wood burn off for 5 to 10 minutes.

Put the potatoes directly on the racks in your pit and set the pan of tomatillos in the pit, then watch your temperature. Any time it drops below 285°F, add a little more wood and charcoal.

Smoke the tomatillos and potatoes for about 1 hour. The potatoes will have darkened from the smoke and softened a bit, and tomatillos should be soft, like water balloons, and have turned a deep, khaki green.

Meanwhile, place the poblanos, one at a time, directly over a gas-stove burner on high heat, turning them with tongs until they are evenly blackened all over, or on a baking sheet under a preheated broiler, again turning as needed until evenly blackened all over. (Or in a pinch, you can even throw them right onto the coals of your pit.) Transfer the poblanos to a bowl, cover with aluminum foil or plastic wrap, and let steam for about 5 minutes. When cool enough to handle, peel the poblanos, remove the stems and seeds (do not rinse the poblanos, or they will lose a lot of their flavor), dice them, and set aside.

When the potatoes and tomatillos are ready, pull them out of the pit. Set the potatoes aside until they are cool enough to handle. Transfer the tomatillos to a stand blender or to a large bowl if using an immersion blender. Place a small skillet over medium-high heat and add 1 tablespoon of the vegetable oil, the árbol chiles, and garlic and fry, stirring frequently, until the garlic is lightly browned and the chiles are toasted but not burned, about 1 minute. Transfer the contents of the skillet to the blender or bowl, season aggressively with salt, and blend until smooth. Taste for seasoning and adjust, if needed. This salsa should be smoky, acidic, spicy, and salty. Set aside to cool.

When the potatoes are cool enough to handle, dice them into about ½-inch pieces. They should be softer than raw yet not completely cooked. But if they are fully cooked, that's okay too.

Now, in a large skillet over medium-high heat, warm 2 tablespoons vegetable oil. Add the chorizo and cook, stirring frequently and allowing it to crumble apart, until cooked through and tender. Depending on the coarseness of the grind, it can take anywhere from 5 to 10 minutes to cook. Using a slotted spoon, transfer the chorizo to a bowl and set aside.

Pour off all but 2 tablespoons of the chorizo fat from the pan and return the skillet to medium-high heat. Add the onion and poblanos, season with salt, and cook, stirring often, until softened, 3 to 4 minutes. Add the potatoes, season with salt and pepper, stir well, and then cook, stirring frequently, until cooked through and soft. The timing will depend on how cooked the potatoes were when they came out of the pit. Return the chorizo to the pan and stir until fully combined and warmed through. Remove from the heat and stir in ¾ cup of the salsa. Taste for seasoning and adjust if needed.

Set a stove-top griddle or large nonstick skillet over medium heat.

Meanwhile, cook the eggs to your liking. I like a soft scramble, but any preparation will work. Mix the eggs into the potato-chorizo hash.

One at a time, warm the tortillas on the griddle or in the skillet just until they are pliable and easy to roll, about 30 seconds on each side. As the tortillas are ready, stack them on one end of a clean kitchen towel and cover with the other end of the towel to keep warm.

Lay the tortillas flat on a work surface. Divide the cheese evenly across all the tortillas. Next, add the hash, dividing it evenly. Then top the hash with a couple spoonfuls of the pico de gallo. Fold in the sides, then fold up the bottom and roll up.

Return the griddle to medium heat and add about 1 tablespoon vegetable oil. When the oil is hot, add two burritos, seam-side down, and cook until crunchy and golden on the underside. Then flip them and cook until crunchy and golden on the second side. Repeat with the remaining burritos, adding more oil as needed.

Serve the burritos immediately, with the remaining tomatillo salsa alongside.

Leftover hash and salsa will keep in separate airtight containers in the refrigerator for up to 6 days.