Trim the pork butt. Remove any thick hard fat from the pork butt, but leave some soft fat on the surface to help keep the meat juicy during the cook.
Add the binder. Mix yellow mustard and apple cider vinegar together, then rub it all over the pork butt.
Make the rub. In a bowl, mix brown sugar, smoked paprika, kosher salt, black pepper, garlic powder, onion powder, chili powder, mustard powder, and optional cayenne pepper.
Season the pork. Coat the pork butt heavily on all sides with the BBQ rub. Let it sit at room temperature for 30 minutes while the smoker preheats, or refrigerate overnight for deeper flavor.
Preheat the smoker. Set the smoker to 250°F using hickory, apple, cherry, pecan, or oak wood.
Start smoking. Place the pork butt directly on the smoker grates with the fat side facing the heat source. Insert a leave-in meat probe into the thickest part of the pork without touching the bone.
Build the bark. Smoke the pork butt until the bark is dark, firm, and set. This usually happens around 160°F to 170°F internal and can take 4 to 6 hours.
Wrap the pork. Place the pork butt on pink butcher paper or heavy-duty foil. Add apple juice, optional brown sugar, and optional butter. Wrap tightly.
Finish smoking. Return the wrapped pork to the smoker and continue cooking at 250°F until the internal temperature reaches about 200°F to 205°F.
Check for tenderness. Start checking around 198°F internal. The pork is done when a probe slides into the meat with very little resistance and the bone wiggles easily.
Rest the pork. Keep the pork wrapped and rest it in a cooler, warm oven, or insulated container for 1 to 2 hours.
Make the finishing sauce. In a small sauce pan, mix apple cider vinegar, ketchup, brown sugar, garlic powder, onion powder, kosher salt, cayenne pepper, salt and heat until everything is dissolved and mixed together. Set aside to call, then transfer to bowl or squeeze bottle
Make the slaw. In a bowl, mix mayonnaise, apple cider vinegar, sugar, Dijon mustard, kosher salt, and black pepper. Add cabbage and carrots, then toss until coated. Refrigerate until ready to serve.
Pull the pork. Transfer the rested pork to a large pan. Remove the bone and any large pieces of unrendered fat. Shred the pork by hand, with forks, or with meat claws.
Season the pulled pork. Mix the bark back into the meat. Add a few spoonfuls of the Carolina Red Sauce and toss until the pork is juicy and flavorful. Add more as needed.
Toast the buns. Butter the cut sides of the buns and toast them on a grill, griddle, or skillet until golden brown.
Build the sandwiches. Pile pulled pork onto the bottom bun. Add BBQ sauce, crunchy slaw, and pickles.
Finish and serve. Add the top bun and serve immediately with extra BBQ sauce, pickles, and napkins.