Heat coals or gas grill for direct heat. Make Best Barbecue Sauce.
Cover and grill chicken, skin sides up, 4 to 6 inches from medium heat 15 minutes. Turn chicken; brush with sauce.
Cover and grill 20 to 40 minutes longer, turning occasionally and brushing 2 or 3 times with sauce, until juice of chicken is no longer pink when centers of thickest pieces are cut.
Heat remaining sauce to boiling; boil 1 minute. Serve sauce with chicken.