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For the pulled pork: Combine the brown sugar, garlic salt, dry mustard, pepper, salt and thyme in a medium bowl. Vigorously rub the pork with the mixture. Wrap the pork in plastic and refrigerate overnight.
Prepare a smoker to 225 degrees F. (Alternatively, preheat an oven to 225 degrees F.) Unwrap the pork and smoke until the center bone is easily removed and the meat is fork-tender, 14 to 16 hours. (If using an oven, roast in a roasting pan covered with foil.) Using 2 forks, break up the pork into bite-size pieces and set aside.
Fill a shallow baking dish large enough to accommodate the apples with 1/2 inch of water. Place the apples in the baking dish, cover with foil and bake until the apples are tender but not mushy, about 15 minutes. Carefully stuff the baked apples with the pulled pork until overflowing. Reserve the remaining pork for another use.