In a large pot combine the brine ingredients. Stir vigorously until the salt has dissolved.
Remove the neck and giblets from both ends of the turkey and reserve in the refrigerator for the gravy. Cut off and reserve the wing tips for the gravy, too. If your turkey has a trussing clamp, leave it in place. Do not truss the turkey. Rinse the turkey inside and out with cold water.
Partially fill a cooler with ice. Open a large, sturdy plastic bag in the cooler. Place the turkey, breast side down, in the bag. Carefully pour the brine over the turkey and then add 3 quarts of cold water. Press the air out of the bag, seal the bag tightly, close the lid of the cooler, and set aside for 18 to 24 hours.
If you are using a charcoal grill, fill a Weber® RapidFire® chimney starter to the rim with charcoal and burn the coals until they are lightly covered with ash. Spread the coals in a half circle or crescent-shaped fire on one side of the charcoal grate. Carefully place a large, disposable drip pan in the center of the charcoal grate and fill it about halfway with warm water. This will help to maintain the temperature of the fire. Put the cooking grate in place, close the lid, and let the coals burn down to low heat. Keep all the vents open.If you are using a gas grill, preheat the grill on high until the temperature reaches 500°F. Then turn the burners down to indirect low heat (300°F to 350°F). Remove the turkey from the bag and rinse it, inside and out, with cold water. Pat dry with paper towels. Discard the brine. Lightly coat the turkey with some of the melted butter. Season with the pepper.
Pour the chicken stock into a sturdy 9 x 13-inch roasting pan. Add the onion, carrots, and celery. Add the reserved turkey neck, giblets, and wing tips. Place the turkey, breast side down, in the roasting pan.
Place the roasting pan in the center of the cooking grate. If you are using a charcoal grill, position the pan so the turkey legs face the charcoal. Drain, then add 2 wood chunks or 2 handfuls of wood chips to the charcoal or put the chunks or chips in the smoker box of a gas grill, following the manufacturer’s instructions. Cook the turkey over indirect low heat with the lid closed for 1 hour.
After 1 hour, to maintain the temperature of your charcoal grill, add 10 to 12 unlit charcoal briquettes to the lit charcoal, using long-handled tongs to tuck the unlit charcoal between the lit charcoal. Drain and add the remaining wood chunks or chips to the charcoal or smoker box. Carefully turn the turkey over in the pan so the breast faces up. Continue to cook the turkey over indirect low heat, with the lid closed, for a second hour.
At the end of the second hour, baste the turkey all over with the remaining butter. If any parts are getting too dark, tightly wrap them with aluminum foil. Once again, add 10 to 12 unlit charcoal briquettes to the lit charcoal to maintain the heat of your charcoal grill. Continue to cook the turkey over indirect low heat. The total cooking time will be 2½ to 3½ hours. The turkey is done when the internal temperature reaches 175°F in the thickest part of the thigh (not touching the bone) and 165°F in the breast. Note: The internal temperature will rise 5°F to 10°F during resting.
Transfer the turkey to a cutting board, loosely cover with foil, and let rest for 20 to 30 minutes before carving. Meanwhile, make the gravy.
Strain the pan liquid through a sieve into a large fat separator and discard all the solids. Add enough chicken stock to equal 4 cups of liquid. Place the roasting pan over a stovetop burner set to medium heat. Add the butter and flour. As the butter melts, stir with a wooden spoon and cook until the mixture turns the color of peanut butter, about 5 minutes, stirring frequently. Add 4 cups of the reserved pan liquid (but not the fat) plus the wine. Bring the gravy to a boil, whisking frequently to dissolve the lumps. Lower the heat and simmer the gravy for a few minutes or until it reaches the consistency you like. If the gravy gets too thick, add more chicken stock a little at a time and simmer until it reaches the right thickness. Turn off the heat. Add the parsley. Add salt and pepper to taste.