Place pork butt in a large pot with a lid. Season with salt and pepper, to taste. Pour broth around pork until it covers about one quarter of the meat. Add liquid smoke. Cover pot and put in oven.
Bake pork until internal temperature registers 190-200°F on an instant-read thermometer and meat is falling apart, 3½-4½ hours. Use a fork to pull at pork, and make sure it comes apart easily. Remove pork butt from oven and let it sit in pot 15-20 minutes. Remove from pot, and discard bone and cooking liquid. Cut fatty parts off meat, then use two forks or your fingers to shred meat into small pieces. Place shredded pork in a large bowl.
Warm barbecue sauce in a small pot over medium heat. Mix shredded meat with barbecue sauce and vinegar until evenly coated. Taste. If pork could use a bit more zing, add more vinegar, to taste. Season with additional salt and pepper.
Pile pulled pork onto rolls, or serve it plain with coleslaw and pickles.