How to Make It
In a large stockpot or plastic tub, mix 1 1/2 gallons of water with the salt and sugar; stir to dissolve the salt and sugar. Add the turkey to the brine, breast side down and refrigerate for 10 to 12 hours.
Remove the turkey from the brine, rinse it in cold water and pat dry with paper towels. Discard the brine.
Light a charcoal grill using 5 pounds of charcoal briquettes. Rake the hot coals into 2 piles on opposite sides of the grill and replace the grill rack. Alternatively, turn the front and back burners of a gas grill to high and preheat to 400°; leave the center burner off. Turn the burners down to medium (350°) just before grilling the turkey.
Meanwhile, place half of the onions, carrot and celery in the turkey cavity. Using kitchen string, tie the turkey legs together, then bring the string around the turkey and tie the wings at the breast. Scatter the remaining onions, carrot and celery in a large, sturdy, disposable roasting pan. Brush the turkey with the melted butter and set it in the roasting pan, breast side down. Pour 2 cups of water into the pan.
Set the turkey on the grill. Cover and cook with the vents open for 1 hour; do not open the grill. (If using a charcoal grill, heat another 1 1/2 pounds of charcoal in a chimney starter after 45 minutes; then remove the pan and rack from the grill and stir the coals; add half of the new coals to each pile and replace the rack and pan.) Turn the turkey breast side up and baste with the pan juices. Continue to grill, covered, for about 1 1/2 hours longer; the turkey is done when an instant-read thermometer inserted in an inner thigh registers 170°.
Transfer the turkey to a carving board, cover loosely with foil and let rest for 20 to 30 minutes. Reserve the pan juices for making the gravy.