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 180 to 200 F on an instant-read thermometer and meat is falling apart, 4 ½ to 5 ½ 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 the pot for 15 to 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.
Notes
If you can find a bone-in pork shoulder, grab that, but a boneless pork shoulder will work fine, too. Don’t be tempted to sub in a pork loin; it’s not fatty enough.
Pork butt doesn't take kindly to blasts of high heat, so don't think you can cheat by turning up the heat and reducing the cooking time. This will yield an unpleasantly tough hunk of meat. Definitely not worth all of your work and patience.
The pork is braised at 300 degrees (don't even bother looking at it until at least 4 ½ hours have passed; you will just be wasting your time) until it reaches an internal temperature of 190 to 200 degrees, or, as Ms. Karmel describes, "until the bone slips right out, clean as a whistle." This temperature will allow the pork to become very tender, and shred easily.