Score both sides of the pork shoulder to allow the marinade seasonings to penetrate the meat.
Rub the dry rub all over the pork shoulder, including inside the slits.
Add apple juice, apple cider vinegar, sugar, hot sauce, and liquid smoke. Transfer the pork shoulder and all the liquid marinade into a plastic bag. Marinate in the refrigerator overnight, or preferably for 24 hours.
Preheat the oven to 250°F (121°C). Transfer the pork shoulder onto a few sheets of heavy-duty aluminum foil and discard the liquid marinade. In a small bowl, combine the dry rub and brown sugar, then rub the mixture all over the pork. Wrap the pork tightly with the aluminum foil, ensuring there are no leaks. Place the pork shoulder in a roasting pan with a rack (preferred) or a deep baking pan, with the aluminum foil opening side up. This will prevent the pork jus from leaking out.
Slow roast the pork shoulder in the oven for 6 hours, as recommended. (If you're pressed for time, you can bake it at 350°F (176°C) for 3 hours.) In the meantime, heat the BBQ sauce in a small saucepan over low heat. Add the brown sugar and stir to mix well. Set aside.
Remove the pork shoulder from the oven and unwrap the aluminum foil. Broil each side for 1 to 2 minutes until slightly charred. Close the foil and let it cool for 30 minutes. Discard the bone and shred the pork with your hands or two forks. Reserve all the pork jus and spoon it over the pulled pork.
How to Make Smoked Pulled Pork?
To smoke the pulled pork, heat a stovetop smoker and add 1 tablespoon of wood chips in the center. Place some pulled pork on aluminum foil, cover it, and place the smoker on a burner over medium heat. Smoke for 5 minutes. Remove from heat and take the pulled pork out of the smoker after 10 minutes.
To serve, drizzle the BBQ sauce generously over the pulled pork. For pulled pork sandwiches, stuff a slider or burger bun with pulled pork, then top with BBQ sauce and coleslaw.