Prepare a charcoal grill: Clean and oil grates, and preheat to medium heat using one chimney of charcoal. The temperature can start as high as 300 degrees and go as low as 250. For best results, coals should be at least 12 inches away from chicken. If necessary, push coals to one side of grill to create indirect heat. Add two large handfuls of soaked pimento (allspice) wood sticks and chips (see note) or other aromatic wood chips to coals, then close grill. When thick white smoke billows from grill, place chicken on grate, skin side up, and cover. Let cook undisturbed for 30 to 35 minutes.