1 tablespoon vegetable oil
4 ounces thick-cut bacon, cut into 1/2-inch pieces
1 tablespoon unsalted butter
5 pounds boneless pork shoulder, cut into 6 pieces
Kosher salt and freshly ground black pepper
3 medium shallots, thinly sliced
1/3 cup Calvados or other apple-flavored brandy
1 1/4 cups apple cider, preferably fresh
2 tablespoons (or more) apple cider vinegar
3 tart, firm apples (such as Pink Lady or Honeycrisp), peeled, cut into 1/2-inch wedges (about 4 cups)
1/2 teaspoon Dijon mustard
Chopped fresh chives
Place a rack in lower third of oven; preheat to 325°. Heat oil in a large heavy pot over medium heat. Add bacon and cook, stirring often, until browned and crisp, 6-8 minutes. Using a slotted spoon, transfer bacon to a paper towel-lined plate and set aside.
Increase heat to medium-high. Add butter to pot with drippings. Season pork shoulder with salt and pepper. Working in batches if needed, cook pork shoulder, reducing heat as needed to prevent overbrowning, until brown on all sides, 8-10 minutes per batch. Transfer pork shoulder to a plate.
Add shallots to pot and cook, stirring often, until shallots begin to soften, about 4 minutes.
Remove pot from heat; add Calvados and stir, scraping up any browned bits from bottom of pot. Return to heat and simmer for 1 minute. Add apple cider and 2 tablespoons vinegar. Bring to a simmer, then return pork shoulder to pot, placing in a single layer on bottom of pot (the meat should not be completely covered).
Cover pot and transfer to oven. Braise pork shoulder, turning after 1 hour, until fork-tender, about 2 hours. Stir in reserved bacon. DO AHEAD: Bacon and pork shoulder can be cooked 2 days ahead. Let cool in braising liquid, uncovered. Chill, uncovered, until cold; cover and keep chilled. Rewarm before continuing.
Using tongs, transfer pork to a deep platter. Skim fat from cooking liquid. Place pot over medium heat and bring liquid to a simmer. Add apples and cook until apples are just tender and sauce is slightly reduced, 8-10 minutes.
Stir Dijon mustard into sauce; season with salt, pepper, and more vinegar, if desired. Pour sauce with apples over pork on platter. Sprinkle with chives.
Serve with Cheesy Grits.