Mix all the ingredients, except the pork, in a glass or plastic bowl until the brown sugar dissolves. The sauce can be made a day ahead and stored, covered, in the refrigerator.
Preheat the grill to 250 degrees F.
Place the pork shoulder on the grill and cook the meat until the internal temperature reaches 180 degrees F, about 10 to 12 hours. Once the pork shoulder is finished cooking, remove from the grill and let rest until it is cool enough to handle.
Strip the skin and bones from the pork shoulder and discard. Pull the pork meat into shreds or chop with a cleaver. Pour half of the sauce into the meat and mix with your hands. Continue adding sauce to the meat until the meat is moist but not soggy.
Serve the pork shoulder with baked sweet potatoes, coleslaw, and some extra barbecue sauce on the side.