For rub, in a large pan stir together first 8 ingredients. Trim excess fat from pork shoulder. Add shoulder to pan and cover with rub. Cover and refrigerate 90 minutes. Meanwhile, soak wood chips for 1 hour in enough water to cover; drain.
For a charcoal grill, prepare grill for indirect grilling with coals arranged around a foil drip pan. Add wood chips to coals. Add grill rack. Test for medium-low heat above the drip pan. Place meat over drip pan. Cover and cook 4 to 5 hours, until a meat thermometer inserted in deepest part of the meat registers 180 degrees F, adding additional coals as needed to maintain temperature.
For a gas grill, start with a full propane tank. Adjust heat for indirect cooking over medium-low heat. (Check manufacturer's directions for your specific grill.) Add soaked wood chips according to manufacturer's directions. Or wrap chips in foil, poke several holes in foil, and add to grill. Place pork on a rack in a roasting pan; set pan on grill rack over the unlit burner. Cover and cook as above.
Meanwhile, prepare Shed Grill-Off Sauce.
Transfer shoulder directly to baking dish to capture juices. With two forks, pull meat into shreds. Remove bone, discard.
To serve, pile 4 ounces meat onto each bun. Drizzle with 3 to 4 tablespoons sauce. Pass remaining sauce. Makes about 10 servings.
Shed Grill-Off Sauce
In a large saucepan stir together brown sugar, tomato sauce, tomato puree, Worcestershire sauce, ketchup, Cajun seasoning, beer, cane syrup or dark corn syrup, soy sauce, mustard, pepper sauce, lemon juice, garlic salt, cayenne pepper, and black pepper, bring to boiling. Reduce heat and keep at a very low simmer for 1 hour, stirring occasionally.