Put the potatoes in a medium saucepan, add water to cover by about 3/4 inch, and add 1 tablespoon salt. Bring to a boil over high heat, reduce the heat to maintain a simmer, and cook until the potatoes are tender but not falling apart, about 5 minutes. Drain the potatoes, transfer to a plate, and set aside.

Heat the oil in a 10-inch straight-sided saute pan over medium-high heat. Add the onion and 1/2 teaspoon salt and cook, stirring occasionally until soft, 5 to 7 minutes. Add the pork and continue to cook until the pork is warm, about 3 minutes. Add the garlic and cook, stirring, until the raw garlic aroma subsides, about 1 minute. Add the potatoes, toss gently to combine, and continue to cook, stirring, until heated through, 1 to 3 minutes more. Season to taste with salt. If the hash is a little dry, add a drizzle of olive oil. Keep warm.

Fill a medium saucepan with 3 inches of water. Add the vinegar and a pinch of salt, and bring the water to a simmer. Crack the eggs one at a time into a small bowl or teacup and then gently slide each egg into the water. Poach the eggs, gently turning once or twice until the whites are completely opaque but the yolks are still soft, 3 to 4 minutes. Using a slotted spoon, remove the eggs from the water and gently blot dry with a towel.

Evenly distribute the hash among 4 plates. Prop a poached egg and a few slices of avocado next to each portion. Sprinkle the egg and avocado with salt. Sprinkle the cilantro and piment d'Espelette (if using) over the hash, and serve immediately.

Combine 2 tablespoons salt and 2 teaspoons pepper in a small bowl and rub the mixture all over the pork. Put the pork, fat side up, in a large roasting pan (about 12x16x3 inches). Cover and refrigerate overnight or for up to 3 days.

Remove the pork from the refrigerator and let sit at room temperature for 1 to 1-1/2 hours before cooking.

Position a rack in the center of the oven and heat the oven to 300 degrees F. Uncover the pork and roast until tender everywhere but the very center when pierced with a fork, 4 to 4-1/2 hours. Add the onion, carrots, garlic, wine, and 1 cup water to the roasting pan and continue to roast, stirring the vegetables occasionally, until the pork is completely tender, about 1 hour more.

Remove the roast from the oven and raise the oven temperature to 375 degrees F. Using tongs, separate the pork into 8 to 10 large, rustic chunks and spread out on the pan. If most of the liquid has evaporated, add a splash more water to the pan to create a little more juice. (It shouldn't be soupy.) Return the pork to the oven and continue to roast until nicely browned on the newly exposed surfaces, about 15 minutes. Remove the pan from the oven, transfer the meat and vegetables to a serving platter, and tent loosely with foil. Let rest for 20 minutes. Skim the excess fat from the juices and serve the juices with the vegetables and meat.