Remove and discard leg truss from turkey. Pull off and discard lumps of fat. Remove giblets and neck (reserve for other uses). Rinse bird well.
In a bowl or pan (at least 12 to 14 qt.), combine 3 quarts water, brown sugar, salt, garlic, peppercorns, and bay leaves. Stir until sugar and salt are dissolved. Add turkey, cover, and chill for 2 hours, turning bird over occasionally.
In a bowl, combine wood chips and 2 to 3 quarts hot water.
Lift turkey from brine and rinse thoroughly under cold running water, rubbing gently to release salt; pat dry with towels. Discard brine. Insert a meat thermometer straight down through the thickest part of the turkey breast to the bone.
On a charcoal barbecue (20 to 22 in. wide) with a lid, mound and ignite 40 charcoal briquets on firegrate. When coals are spotted with gray ash, in about 20 minutes, push equal portions to opposite sides of firegrate. Place a foil drip pan between mounds of coals. To each mound, add 5 briquets and 1/2 cup drained soaked wood chips now and every 30 minutes (until all chips are used). Set grill in place. Set turkey, breast up, on grill over drip pan. Cover barbecue and open vents.
On a gas barbecue (with at least 11 in. between indirect-heat burners), place 1 cup drained soaked wood chips in the metal smoking box or in a foil pan directly on heat in a corner. Turn heat to high, close lid, and heat for about 10 minutes. Adjust gas for indirect cooking (heat parallel to sides of bird and not beneath) and set a metal or foil drip pan in center (not over direct heat). Set grill in place. Set turkey, breast up, on grill over drip pan. Close barbecue lid. Add another cup of wood chips (sprinkle through or lift grill) every 30 minutes until all are used. If edges of turkey close to heat begin to get too dark, slide folded strips of foil between bird and grill. Fat in drippings may flare when barbecue lid is opened; quench by pouring a little water into the pan.
Cook turkey until thermometer registers 160°, in 2 to 3 hours; start checking after 1 hour.
Drain juices from cavity into drippings and reserve for other uses. Transfer turkey to a large platter; let rest 15 to 30 minutes before carving.
Nutritional analysis per 1/4 pound boned cooked turkey with skin, based on percentages of white and dark meat in an average bird.