Preheat your oven to 325°F. Pat the 3 ½ lb bone-in pork shoulder also called pork butt completely dry with paper towels — this is key for a good sear. Season all over generously with 1 ½ tsp kosher salt and 1 tsp black pepper freshly ground.

Heat the 2 tbsp olive oil in a large Dutch oven over medium-high heat until shimmering. Add the 3 ½ lb bone-in pork shoulder also called pork butt and sear undisturbed for 4–5 minutes per side until deeply golden-brown on all sides, including the ends. Don't rush this step — the crust adds major flavor. Transfer the 3 ½ lb bone-in pork shoulder also called pork butt to a plate and set aside.

Reduce the heat to medium. Add the sliced yellow onion halved and sliced into half-moons, 3 carrots peeled and cut into 2-inch chunks, and 3 stalks celery cut into 2-inch pieces to the Dutch oven. Cook, stirring occasionally, until softened and lightly browned, about 5–6 minutes. Scrape up any browned bits from the bottom as you stir.

Add the smashed 4 cloves garlic smashed and 2 tbsp tomato paste. Stir and cook for 1–2 minutes until the 2 tbsp tomato paste darkens slightly and becomes fragrant.

Pour in the 2 cups hard apple cider dry and 1 cup chicken stock low-sodium. Stir in the 2 tbsp whole grain mustard. Bring to a gentle simmer and scrape up any remaining browned bits from the bottom of the pot.

Nestle the seared 3 ½ lb bone-in pork shoulder also called pork butt back into the pot — it should be about halfway submerged in liquid. Tuck in the 5 pieces fresh thyme sprigs sprigs, 2 pieces bay leaves, and 2 cups hard apple cider dry wedges (if using) around the 3 ½ lb bone-in pork shoulder also called pork butt.

Cover the Dutch oven with a tight-fitting lid and transfer to the preheated oven. Braise for 2.5 to 3 hours, turning the 3 ½ lb bone-in pork shoulder also called pork butt once halfway through, until the meat is completely fork-tender and pulls apart easily. A fork should slide in with almost no resistance.

Carefully transfer the 3 ½ lb bone-in pork shoulder also called pork butt to a cutting board and tent loosely with aluminum foil to rest. Discard the 5 pieces fresh thyme sprigs sprigs and 2 pieces bay leaves. Using a ladle or large wooden spoon, skim off any excess fat from the surface of the braising liquid.

Place the Dutch oven on the stovetop over medium-high heat. Simmer the braising liquid and vegetables for 8–10 minutes until slightly reduced and the sauce coats a wooden spoon. Remove from heat and swirl in the cold 1 tbsp unsalted butter cold until melted and glossy. Add the 1 tsp apple cider vinegar and taste for 1 ½ tsp kosher salt, adjusting as needed.

Use two forks or your hands to pull the 3 ½ lb bone-in pork shoulder also called pork butt into large, rustic chunks or shred it completely — your choice. Arrange on a serving platter or individual plates, spoon the pan sauce and vegetables generously over the top, and garnish with fresh fresh flat-leaf parsley roughly chopped if desired. Serve immediately.