Prepare the Pork Shoulder : Rub olive oil all over the pork. Mix smoked paprika, garlic powder, onion powder, salt, black pepper, and cayenne in a small bowl. Rub the spice mixture evenly over the pork.
Cook the Pulled Pork : Slow Cooker Method: Place pork in slow cooker, add chicken broth. Cook on low for 6–8 hours until pork is tender and easily shredded. Oven Method: Preheat oven to 325°F (160°C). Place pork in a roasting pan, add chicken broth, cover with foil, and bake 3–4 hours until tender.
Shred the Pork : Remove pork from slow cooker or oven. Use two forks to shred the meat. Stir in 1 cup BBQ sauce (or more to taste).
Prepare the Coleslaw : In a large bowl, combine shredded cabbage and carrots. Add mayonnaise, apple cider vinegar, honey, salt, and pepper. Mix until well combined.
Assemble the Sandwiches : Slice buns and lightly toast if desired. Add a generous portion of pulled pork on each bun. Top with a layer of coleslaw. Drizzle extra BBQ sauce if desired.
Serve : Serve sandwiches immediately with chips, pickles, or baked beans.