This is it—the ultimate pork shoulder. My inspiration is no less than the pulled pork sandwich at Lexington Barbecue (formerly the Honey Monk) in Lexington, North Carolina.
Step 1: Make the rub: Place the paprika, brown sugar, salt, black pepper, white pepper, mustard, garlic powder, and cayenne in a small bowl and stir to mix. (Actually, if you don’t have sensitive skin, your fingers work better for mixing a rub than a spoon or whisk does.) Set aside 1-1/2 tablespoons of the rub for the Lexington Vinegar Sauce. Sprinkle the remaining rub all over the pork, patting it onto the meat with your fingertips.
Step 2: Set up the grill for indirect grilling and preheat to medium-low. If using a gas grill, place all of the wood chips or chunks in the smoker box or in a smoker pouch and run the grill on high until you see smoke, then reduce the heat to medium-low.If using a charcoal grill, place a large drip pan in the center, preheat the grill to medium-low, then toss 1 cup of the wood chips or chunks on the coals.
Step 3: When ready to cook, place the pork, skin side up, if there is one, in the center of the hot grate, over the drip pan and away from the heat, and cover the grill. Cook the pork until darkly browned on the outside and very tender inside, 4 to 6 hours. To test for doneness, use an instant-read meat thermometer: The internal temperature of the pork should be about 195 degrees. (Yes, this is very well-done—that’s how you get the pork tender enough to pull.) If the pork starts to brown too much (and it probably will), cover it loosely with aluminum foil, but remember that those browned bits are good, too. If using a charcoal grill, every hour you’ll need to add 12 fresh coals and 1/2 cup of wood chips or chunks to each side.
Step 4: Transfer the cooked pork to a cutting board, cover it loosely with aluminum foil, let it rest for 20 minutes, then pull the pork into large pieces, discarding any bones or lumps of fat (you’ll probably want to wear latex gloves or even heavy-duty insulated rubber gloves to do this). Using your fingertips or a fork, pull each piece of pork into thin shreds. Or use a cleaver to finely chop it. Transfer the pulled pork to a large aluminum foil pan and stir in 1 to 1-1/2 cups of the Lexington Vinegar Sauce—enough to keep the meat moist. If you are not quite ready to serve, cover the pan with aluminum foil and place it on a warm—not hot—grill or in an oven turned on low to keep warm.
Step 5: To serve, have your guests mound the pulled pork onto hamburger buns. Top with Lexington Slaw and let everyone add additional vinegar sauce to taste.