In a small bowl, mix together brown sugar, paprika, garlic powder, onion powder, cayenne pepper, salt, and pepper to create a dry rub. Rub the dry rub all over the pork shoulder, ensuring it is well-coated. Let it marinate for at least 30 minutes to allow the flavors to penetrate the meat.
Preheat your grill or smoker to 225-250°F (107-121°C). Sear the pork on all sides over high heat for a few minutes to develop a flavorful crust.
Once seared, move the pork to the indirect heat side of the grill or smoker. Pour the chicken or vegetable broth into the bottom of the pan to keep the meat moist during cooking. Cook the pork low and slow for 5-7 hours or until the internal temperature reaches at least 195°F (90°C). The meat should be tender and easily shred with a fork.
Once cooked, let the pork rest for a few minutes. Use two forks to shred the meat. Discard any excess fat.
In a saucepan, combine the barbecue sauce and apple cider vinegar. Heat over medium heat until warmed through.
Mix the shredded pork with the warm barbecue sauce mixture. Toast the hamburger buns or rolls on the grill or in an oven. Spoon the pulled pork onto the buns and top with coleslaw if desired. Serve with additional barbecue sauce on the side.