Dissolve the salt and sugar in 2 gallons cold water in a large (at least 16 quart) stockpot or a clean bucket. (You can also add fresh herbs like rosemary, thyme or bay leaf, or aromatics like onions and garlic.) Add the turkey, cover and refrigerate, or set in a very cold place (that stays somewhere between 32 to 40 degrees F) for at least 12 hours and up to 24 hours.
Remove the turkey from the brine 1 hour before grilling. Rinse well under cold water and pat dry to remove the salt.
Prepare a charcoal grill for indirect grilling. The temperature inside the grill should be 350 degrees F.
Place the turkey on the grill, cover and grill until an instant-read thermometer inserted in the thigh registers 175 degrees F, 2 hours 15 minutes to 2 1/2 hours. Let the turkey rest 1 hour before carving.