1 cup Dijon mustard
2 tablespoons dark brown sugar
2 tablespoons kosher salt
2 tablespoons freshly ground pepper
1 tablespoon sweet paprika
1 tablespoon onion powder
One 12- to 14-pound bone-in pork shoulder, with skin
About 50 hardwood charcoal briquettes
8 cups small hardwood smoking chips, soaked in water for 30 minutes and drained
Eastern Carolina Sauce for serving
Preheat the oven to 275. In a medium bowl, whisk the mustard with the brown sugar, salt, pepper, paprika and onion powder.
Set the pork shoulder, fat side up, in doubled 14-by-18-inch disposable aluminum roasting pans. Brush the pork with the mustard mixture. Roast, uncovered, for 12 hours, until the meat is very tender and is pulling away from the shoulder bone.
Tilt the pan and pour the roasting juices into a medium bowl; you should have about 1 1/4 cups. Refrigerate the juices for 30 minutes. Skim off the fat before using.
Meanwhile, light 10 of the charcoal briquettes. When the coals are hot, cover them with the remaining 40 briquettes. When all the coals are hot, arrange 6 cups of the soaked wood chips around the coals. Set the roasting pan on the grill grate over the coals and wood chips. Cover the grill, partially open the air vents and smoke the pork shoulder for 30 minutes.
Carefully remove the pork and the grill grate and stir the coals a few times. Scatter the remaining 2 cups of soaked wood chips over the coals. Replace the grill grate and return the pork to the grill. Cover and smoke for 30 minutes longer.
Transfer the pork to a work surface and let rest for 30 minutes. Pull the meat off of the bones; discard the bones, gristle, skin and fat. Using tongs and a fork, or your fingers, finely shred the meat and transfer it to a large bowl. Toss the shredded meat with 1/4 cup of the reserved roasting juices and serve with Eastern Carolina barbecue sauce.