Preheat oven to 325 degrees F.
Rub the chicken thoroughly in the spice rub. In a large Dutch oven, whisk the beer, cider vinegar, ketchup, liquid smoke, and Worcestershire sauce until combined. Nestle the chicken into the Dutch oven, overlapping if necessary, and bring the mixture to a boil.
Cover the pot and move to the preheated oven. Cook the chicken until it is falling off the bone, 75 to 90 minutes. Remove chicken with tongs to a large sheet pan or bowl. Allow to cool a few minutes, then shred using two forks (or my favorite bear claws). Discard any bones and gristle. Place the shredded chicken back into the cooking liquid and cool completely. (This step can be made up to two days in advance; just cover and refrigerate until ready to move forward.)
Turn the oven on broil. Push the chicken through a fine mesh sieve to remove excess sauce (do not discard the cooking liquid). Arrange the chicken on two sheet pans and broil (working in batches) until the chicken is crisp and some of the ends are blackened, 6 to 10 minutes each.
Toss the pulled chicken with your favorite barbecue sauce, a few tablespoons at a time, to your taste. Add a few tablespoons of cooking liquid to moisten, if desired. (I always save some liquid to add to the meat when reheating.) Serve barbecue chicken with steamed potato buns and vinegar slaw.
For the vinegar slaw, combine the coleslaw mix with vinegar, sugar, salt, and pepper. Refrigerate for at least one hour before serving. Taste, and adjust any seasonings if desired.
For the southern spice rub, mix all ingredients until well-blended. Refrigerate excess for later use.
NotesCook's Note: Don't use chicken breasts for this recipe because the meat will become very dry under the heat of the broiler. The fat in the dark meat will help keep the chicken moist.