Dissolve the salt and sugar in 5 cups water in a large bowl and add the chicken to it. If needed, add water to cover the chicken, and let it sit for 2 to 3 hours in the refrigerator.
Heat a charcoal or gas grill, keeping part of the grill cool for indirect grilling. Remove the chicken from the brine, pat dry, and season with salt and pepper. Put the chicken pieces skin side down on the hot part of the grill and cook until lightly browned, 3 to 5 minutes. Transfer the chicken to the cool side of the grill, close the lid, and cook for 50 to 60 minutes, turning once.
Uncover the grill. Gradually brush the chicken with the barbecue sauce as you continue to cook it, turning occasionally, until a meat thermometer inserted into the thickest part of one of the pieces reads 165°F, 20 to 25 minutes. Serve hot, warm, or at room temperature. (Store leftover barbecued chicken wrapped in foil in the fridge for up to a few days.)
