Bring 2 gallons of water to boil in a large stockpot. Stir in salt and sugar until completely dissolved. Remove from the heat and add the diced carrots, onion, celery, leeks, spices and fresh herbs. Place the stockpot in refrigerator until cold.
Remove the giblets from the turkey. Add the turkey to the stockpot and place a plate on top to weigh the turkey down to insure full immersion. Refrigerate the whole turkey for 72 hours. Remove from brine and place the turkey in a large, disposable, aluminum-roasting pan. Allow the turkey to come to room temperature.
Meanwhile, start the grill. When the coals are white hot, bank them up high along the perimeter of the grill so they form a circle. Place the roasting pan on the grill rack and cover with the lid. Control the temperature of the grill by opening and closing the vents. If the grill is too hot, close the vents slightly, but not all of the way. If the grill is too cold, open the vents. While the turkey is cooking, baste frequently with olive oil or butter and pan juices. It may be necessary to add more coals during the cooking process. If so, fire the coals in a cone separately and add them when they are hot.
It's important to know that brined turkeys cook more rapidly than un-brined turkeys. So you should check the internal temperature of the turkey after 1 1/2 hours grilling time. Remove the turkey from the grill when the deepest part of the thigh reaches 160 F. Let the turkey rest for 20 minutes before carving.