Position an oven rack 6 inches from the broiler element and heat the broiler to high. Toss the chiles with the oil, 1/4 teaspoon of the salt, and a few generous grinds of pepper. Set on a rimmed baking sheet lined with aluminum foil and broil the chiles, flipping every 1 to 2 minutes, until they brown, blister, and blacken all over, about 5 minutes total. Remove the chiles from the oven, wrap in the foil, and let cool to room temperature. Then peel off the skin and core and seed them. Cut them into 1/2-inch pieces.

Cook the bacon in a large Dutch oven over medium heat until it browns and renders much of its fat, about 6 minutes. Using a slotted spoon, transfer to a plate lined with paper towels. Add the onion and garlic to the pot, sprinkle with the remaining 1/2 teaspoon salt, and cook, stirring, until they soften and brown in places, about 6 minutes. Add the cumin and cook, stirring, for 30 seconds.

Raise the heat to high, add the beer, and cook, stirring to pick up any browned bits on the bottom of the pot, until it has almost evaporated, 4 to 6 minutes. Add the chicken broth, potatoes, half of the cilantro, the oregano, bay leaf, and the chiles and bring to a boil. Reduce to a gentle simmer, cover, and cook until the potatoes are just barely tender, about 15 minutes. Stir in the pork and vinegar and cook until the potatoes are completely tender, about 10 more minutes. Season to taste with more salt, pepper, and vinegar. Serve, sprinkled with the bacon, cheese, and the remaining cilantro.

Combine 2 cups of the apple cider or juice with the salt, brown sugar, garlic, and thyme in a 3- to 4-quart saucepan and bring to a boil over high heat, stirring so the salt and sugar dissolve, about 3 minutes. Add the remaining apple cider or juice and cool to room temperature. Transfer to a large container, add the pork, cover, and refrigerate for at least 8 hours and up to 18 hours.

Position a rack in the center of the oven and heat the oven to 450 degrees F. In a small bowl, mix the maple syrup, mustard, thyme, and pepper. Drain the pork and pat dry with paper towels. Brush the pork all over with the mustard mixture.

In a medium bowl, toss the fennel and apple with the oil, salt, and a few generous grinds of pepper. Scatter the mixture in the bottom of a large roasting pan (large enough to hold the pork with a couple of inches of space around the perimeter). Put the pork, fat side up, on top of the fennel and apples. Roast the pork until the crust just starts to brown, about 15 minutes. Reduce the heat to 350 degrees F and continue cooking until an instant-read thermometer inserted into the thickest part of the loin registers 145 degrees F, 30 to 50 minutes more.

Let rest for 10 minutes and then thinly slice a quarter to a third of the pork. Serve, topped with the fennel, apple, and juices. Allow the remaining pork to cool to room temperature, wrap well with foil, and refrigerate for up to 5 days.