Combine brown sugar, smoked paprika, salt, black pepper, garlic powder, onion powder, cayenne, mustard powder, and cumin in a small bowl. Mix thoroughly and set aside.
Pat the pork shoulder completely dry with paper towels. Trim the fat cap to about 1/4 inch thick and remove any large, loose pieces of fat from the sides.
Rub the entire surface with yellow mustard. Generously coat all sides with the dry rub, pressing firmly to adhere. Refrigerate uncovered on a wire rack for 8-24 hours for best results.
Set up your smoker for indirect cooking at 225°F to 250°F (107-121°C). For charcoal, use a two-zone fire; for pellet, fill the hopper and preheat; for a kettle, use the snake method.
Place the pork fat-side up on the smoker grate, away from direct heat. Insert a leave-in probe thermometer into the thickest part, avoiding the bone. Close the lid and cook undisturbed for the first 3 hours.
Combine apple cider vinegar and apple juice in a spray bottle. After 3 hours, lightly spritz the pork every 45-60 minutes, working quickly to minimize heat loss. Add wood chunks as needed to maintain steady smoke.
When the internal temperature stalls around 150-170°F (66-77°C), maintain your smoker temperature and wait it out. To speed things up, wrap the pork in pink butcher paper or foil at about 165°F (74°C) to push through the stall.
After wrapping, continue cooking until the internal temperature reaches 200-205°F (93-96°C) and a probe slides in with zero resistance, like butter. The bone should wiggle freely when done.
Remove the pork from the smoker and rest it wrapped in an insulated cooler for 2-4 hours, or on a cutting board for a minimum of 30-45 minutes to allow the juices to redistribute.
Carefully unwrap the pork over a pan to catch the juices. Remove the bone, discard any unrendered fat, and shred the meat using claws or forks. Pour the reserved juices and BBQ sauce over the pork, toss to combine, and serve.