In a small mixing bowl, combine the 2 tbsp brown sugar, 1 tbsp smoked paprika, 1 tbsp kosher salt, 1 tsp black pepper, 1 tsp garlic powder, 1 tsp onion powder, 1 tsp dried mustard, and ½ tsp cayenne pepper. Mix well until evenly blended.

Pat the 4 lb bone-in pork shoulder completely dry with paper towels. This helps the rub stick and promotes a better crust. Rub the spice mixture all over every surface of the bone-in pork shoulder, pressing it in firmly. For best results, do this the night before and refrigerate uncovered — but even 30 minutes at room temperature makes a difference.

Preheat your oven to 300°F. Heat the 1 tbsp vegetable oil in a large Dutch oven over medium-high heat. Once shimmering, sear the 4 lb bone-in pork shoulder on all sides until deeply browned, about 3–4 minutes per side. Don't rush this step — the sear builds flavor. Remove the 4 lb bone-in pork shoulder and set aside.

Pour the ½ cup apple cider vinegar, ½ cup chicken broth, and 2 tbsp Worcestershire sauce into the Dutch oven. Scrape up any browned bits from the bottom with a wooden spoon — that's pure flavor. Return the seared 4 lb bone-in pork shoulder to the pot, fat-side up. Spoon half the BBQ sauce over the top of the bone-in pork shoulder.

Cover the Dutch oven tightly with its lid and transfer to the preheated oven. Braise for 7–8 hours, until the 4 lb bone-in pork shoulder is completely fall-apart tender and an internal thermometer reads at least 200°F. The bone should wiggle freely and pull out cleanly.

Carefully remove the 4 lb bone-in pork shoulder from the Dutch oven and place on a large cutting board. Let it rest for 15 minutes — this makes it easier to handle and juicier. Using two forks (or your hands with heat-safe gloves), shred the meat, discarding any large pieces of fat or the bone.

Skim any excess fat from the surface of the braising liquid in the pot. Return the shredded 4 lb bone-in pork shoulder to the pot and stir in the remaining half cup of grill sauce. Toss everything together so the 4 lb bone-in pork shoulder is coated and soaks up the juices. Taste and adjust seasoning with 1 tbsp kosher salt if needed.

While the 4 lb bone-in pork shoulder rests, make the coleslaw. In a large mixing bowl, whisk together the ½ cup mayonnaise, ½ cup apple cider vinegar, 1 tbsp sugar, 1 tsp celery seed, 1 tbsp kosher salt, and 1 tsp black pepper until smooth.

Add the shredded 4 cups green cabbage finely shredded, 1 cup red cabbage finely shredded, and 1 cup carrots to the dressing. Toss well until everything is evenly coated. Taste and adjust seasoning. Refrigerate for at least 15 minutes before serving so the flavors meld and the slaw softens slightly.

Toast the 8 brioche hamburger buns cut-side down in a dry skillet or under the broiler for 1–2 minutes until lightly golden. Pile a generous heap of pulled 4 lb bone-in pork shoulder onto the bottom bun, top with a big scoop of coleslaw, add 8 slices dill pickle chips if using, and finish with extra 1 cup BBQ sauce. Add the top bun and serve immediately.