Prep. To create the braising liquid, combine the chicken broth, apple juice, BBQ sauce, apple cider vinegar, and BBQ dry rub in a bowl and blend well.
Trim all excess fat from the pork butt.
Lightly coat the pork butt with mayonnaise. Season the pork butt liberally with the remaining BBQ dry rub.
Stand an apple upright on a cutting board. Slice one side off of the apple, approximately 1/2-inch from the center. Do the same with the opposite side of the apple. Repeat with the remaining apples. Enjoy the ends of the apples as a fresh snack.
Cook. Set the apples on the bottom of the slow cooker with one cut side down to create a base for the pork butt. This will keep the pork butt out of the braising liquid during the beginning of the cooking process so that the dry rub has time to penetrate the meat.
Add enough of the braising liquid to the slow cooker so that all but the upper 1/4-inch of the apples is immersed. Reserve the remaining liquid for adding to the finished pulled pork.
Set the pork butt on the apples.
Cover the slow cooker and cook the pork butt until it reaches an internal temperature of 190°F, approximately 1 hour per pound.
Remove the pork butt from the crockpot and place in a large casserole dish or aluminum pan. Remove the cooking liquid from the slow cooker and discard. Alternatively, you can reserve the liquid for use as the base of a flavorful sauce, though you will want to skim as much fat as possible off of the top once it has cooled, as it will be very oily from the rendered pork fat.
Use forks or bear paws to shred the pork butt. Add the remaining braising liquid to the pulled pork and mix the pork well. If you are not serving the pulled pork immediately, then return it to the slow cooker, cover, and set the dial to warm. Serve. To serve, place the pulled pork on buns and serve with additional BBQ sauce.