Remove the pork shoulder from the refrigerator and let it rest at room temperature for at least 30 minutes. This helps it cook more evenly.
In a small bowl, combine the salt, pepper, garlic powder, smoked paprika, and brown sugar. This will be your dry rub.
Rub the pork shoulder with olive oil, then generously sprinkle with the spice mixture. Massage it into the meat to ensure it's fully coated.
Preheat your grill or oven to 250°F (120°C). If using a charcoal grill, place the meat over indirect heat to cook slowly.
Place the pork shoulder on the grill or in a roasting pan, then cover it with aluminum foil. Cook at a low temperature for about 4 hours, checking it occasionally.
Mix together the apple cider vinegar and Worcestershire sauce, then baste the meat with it every hour to keep it moist.
During the last hour of cooking, remove the foil and brush generously with BBQ sauce. Let it cook until the sauce caramelizes slightly.
Remove the pork shoulder from the oven or grill and let it rest for at least 20 minutes. Then, shred it with a fork and mix with additional BBQ sauce.
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