Set up grill for indirect grilling. Rinse chicken inside and out and dry thoroughly. Generously season inside and out with coarse salt and ground pepper. Clean and lightly oil hot grates.
Place chicken, breast side up, on opposite side of grill from heat. Cover and grill until cooked through, or a thermometer inserted in the thickest part of a thigh (avoiding bone) registers 160 degrees, about 1 1/2 hours, rotating halfway through cooking (if using a charcoal grill, add a couple of handfuls of charcoal every half hour to maintain heat).
About 15 minutes before chicken is done, brush with 1/2 cup barbecue sauce. Let chicken rest 20 minutes before carving.