Dissolve salt in 2 quarts cold water in large container. Submerge chicken, cover with plastic wrap, and refrigerate 1 hour. Rinse chicken under cold water and dry thoroughly with paper towels. Season chicken with pepper.

Turn all burners to high and heat with lid down until very hot, about 15 minutes. Scrape grate clean with grill brush. Dip wad of paper towels in oil; holding wad with tongs, wipe cooking grate. Leave primary burner on high and turn off other burner(s).

Cook chicken on all sides over hotter part of grill until skin is lightly browned and meat has faint grill marks, 6 to 8 minutes. (If constant flare-ups occur, slide chicken to cooler side of grill and mist fire with water from spray bottle.)

Move chicken, skin-side down, to cooler side of grill, with thicker side of breast facing coals. Cover loosely with aluminum foil, cover grill, and continue to cook until instant-read thermometer inserted into thickest part of breast registers 150°, 15 to 25 minutes longer.

Brush bone side of chicken with sauce, if using. Move chicken, bone-side down, to hotter side of grill and cook until browned, 4 to 6 minutes. Brush skin side of chicken with sauce; turn chicken over and continue to cook until browned and instant-read thermometer inserted into thickest part of breast registers 160°, 2 to 3 minutes longer. Transfer chicken to plate and let rest, tented with foil, 5 minutes. Serve, passing remaining sauce separately.