Braised Pork Shoulder with Chorizo and White Beans

Trim the excess fat from the pork shoulder. Cut the meat away from the bone, if necessary. Roll the meat into a bundle and tie with butcher’s twine to secure.
In a stockpot, combine the salt, sugar, peppercorns, chile flakes, 2 fresh or 1 dried bay leaves, 8 sprigs thyme, and the quartered onion with the water and bring to a boil over high heat. Remove the pot from the heat and let the brine cool completely.
Pour the brine into a large bowl. Submerge the pork shoulder in the brine, cover, and refrigerate overnight.

Preheat the oven to 325°F (163°C).
Remove the pork shoulder from the brine and rub it down with the paprika. (No need to wipe it off first.)
In a large Dutch oven, lidded pot, or roasting pan, combine the pork shoulder, carrot, fennel, sliced onion, garlic, a bunch of thyme, and 1 bay leaf. Pick a cooking vessel that is big enough to hold everything, but not so big you need a gallon of stock to reach three-quarters of the way up the shoulder to keep the pork moist as it cooks.
In a small bowl, whisk together the tomato paste, wine, honey, and vinegar. Pour the mixture over the pork. Add enough stock to come up to the “shoulders” of the meat. Cover the pork tightly with the lid or seal with two pieces of foil.
Cook until the pork is very tender but not totally falling apart, 4 to 6 hours. Remove the bay leaves and thyme branches and discard. The pork can be braised up to 1 day ahead. Cool and refrigerate. Reheat in a 350°F (180°C) oven for 45 minutes.

While the pork is braising, in a medium saucepan over medium heat, warm the oil. Add the chorizo, onion, apples, garlic, and paprika. Cook until the vegetables are soft but not browning, 8 to 10 minutes.
Increase the heat to medium-high. When the onion starts to color, pour in the cider. Cook until the cider reduces almost entirely, 2 to 4 minutes.
If using dried soaked beans, add the soaked beans to the pot with enough water to cover by 1-inch (25 mm). Cook over medium heat until the beans are tender, 1 to 1 1/2 hours.
If using canned beans, dump in the beans and liquid from the cans and cook until the liquid reduces a bit and the flavors meld, 15 to 20 minutes.

Add the beans to the pot with the pork and continue cooking, uncovered, until the beans and cooking liquid are bubbling and a little reduced, about 20 minutes more. Taste and season with salt, if needed. Use two forks to gently pull the pork apart into large chunks, discarding any twine.
In a medium bowl, toss together the apple, parsley, lemon juice, oil, and 1/4 teaspoon salt. Top the pork and beans with the apple salad and serve out of the pot.