In a small bowl, combine the barbecue seasoning, garlic, pepper, and salt. If using pork butt, chip it into thick strips or chunks, about the size of a small fist. Place the pork in a large disposable aluminum foil pan. Coat the pork on all sides with Stubb’s Bar-B-Q Spice Rub.
Prepare a grill for indirect cooking. For a charcoal grill, when the coals are ashed over, rake or spread them out in one part of the grill so the food can cook to the side and not directly over the coals. (For a gas grill, fire up the burners on one part of the grill, so the food can cook to the side of the heat but not directly over it.) Cover the grill and bring it to around 250°F. Drain and add half of the wood (if using) for smoking.
Set the pan of pork over indirect heat and cook in the covered grill for about 3 hours. After the first 10 minutes of cooking, baste the pork with Moppin' Sauce. To ensure even cooking, set an over thermometer next to the pan, also over indirect heat. Every hour, check it, baste it with the Moppin' Sauce and add more fuel and wood as needed to maintain a smoky 250°F heat. The pork is done when the pieces are browned on the outside but tender on the inside. When the meat reaches an internal temperature of 165°F to 175°F, it’s cooked enough to be sliced.