Pulled Pork Sandwich with Coleslaw

Method

1. Pat the pork shoulder dry with paper towels, then trim off any thick fat cap down to about 1/4 inch.
2. In a small bowl, mix the brown sugar, paprika, salt, garlic powder, onion powder, black pepper, and cayenne to make a dry rub, then coat the pork shoulder all over, pressing it into the meat.
3. Set up your grill for indirect heat by lighting charcoal or burners on only one side, aiming to hold a steady temperature of 250F, and place a drip pan under the empty side.
4. Place the pork shoulder fat-side up on the cool side of the grill, away from direct flame, close the lid, and cook for about 5 hours, maintaining 250F and adding fuel as needed.
5. Every hour, spritz the pork lightly with the 1 cup of apple cider vinegar to keep the surface moist and help build a good bark.
6. The pork is done when a meat thermometer inserted into the thickest part reads at least 195F and slides in with almost no resistance, feeling like inserting it into soft butter.
7. Remove the pork from the grill, wrap it loosely in foil, and let it rest for 20 to 30 minutes so the juices redistribute before shredding.
8. While the pork rests, make the coleslaw by whisking together the mayonnaise, vinegar, and sugar in a large bowl, then tossing with the shredded cabbage until evenly coated; season with a pinch of salt.
9. Unwrap the pork and shred it into bite-sized pieces using two forks, discarding any large pieces of unrendered fat.
10. Mix about 1 cup of barbecue sauce into the shredded pork, tasting and adding more to your preference.
11. Toast the hamburger buns lightly, then pile a generous scoop of saucy pulled pork onto each bottom bun.
12. Top each sandwich with a big spoonful of coleslaw directly on the meat and close with the bun lid.