In a small bowl, combine smoked paprika, brown sugar, kosher salt, black pepper, garlic powder, onion powder, cumin, cayenne, and mustard powder. Mix well.

Pat the pork shoulder completely dry with paper towels. Coat the entire surface generously with the dry rub, pressing it firmly into every side and crevice. For best results, wrap the seasoned pork in plastic wrap and refrigerate overnight or for at least 2 hours.

Preheat your oven to 300°F (150°C). Heat a large Dutch oven or heavy-bottomed oven-safe pot over high heat with a drizzle of oil. Sear the pork shoulder on all sides until a deep, dark crust forms, about 3–4 minutes per side. This step builds enormous flavor in the bark.

Remove the pork and set aside briefly.

Pour the apple cider vinegar, chicken broth, Worcestershire sauce, and liquid smoke (if using) into the bottom of the Dutch oven. Return the seared pork shoulder to the pot, fat cap facing up.

Cover tightly with a lid or two layers of heavy-duty foil and transfer to the oven. Cook at 300°F for 7–8 hours, or until the pork is fall-apart tender and registers 195–205°F on an instant-read thermometer.

Carefully transfer the pork to a large cutting board or baking sheet. Let it rest uncovered for 20–30 minutes. Reserve the pan drippings.

Using two forks or your hands (with heat-resistant gloves), shred the pork into long strands, discarding any large pieces of fat or the bone.

Skim the fat from the reserved drippings. Add 2–3 tablespoons of the skimmed drippings back to the shredded pork along with the BBQ sauce. Toss to coat everything evenly. Taste and adjust seasoning with salt, pepper, or an extra splash of apple cider vinegar.

Melt butter in a skillet over medium heat and toast the brioche buns cut-side down until golden and slightly crispy, about 1–2 minutes.

Pile a generous heap of saucy pulled pork onto the bottom bun. Top with a scoop of creamy coleslaw, a few dill pickle slices, and extra BBQ sauce if desired. Close the sandwich, press gently, and serve immediately.