In a large pot combine the brine ingredients. Stir vigorously until the salt is dissolved.
Remove the neck and giblets from 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. If the turkey has a pop-up thermometer, discard it.
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 the 3 quarts/2.8 liters of cold water. The turkey should be almost completely submerged. If some the back is exposed above the brine, that’s okay. 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 using wood chips, soak in water for at least 30 minutes (no need to soak wood chunks).
Fill a 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 (250° to 350°F/120º to 180ºC). Keep all the vents open.
Remove the turkey from the bag and rinse it, inside and outside, with cold water. Pat dry with paper/kitchen towels. Discard the brine. Lightly coat the turkey with some of the melted butter. Season with the pepper.
Place one foil pan inside the other and pour the chicken stock into the top pan. Add the onion, carrots, and celery. Add the reserved turkey neck, giblets, and wing tips. Place the turkey, breast side down, in the foil pan.
Add two wood chunks or drain and add two handfuls of wood chips to the charcoal, and close the lid. When the wood begins to smoke, place the pan in the center of the cooking grate. Position the pan so the turkey legs face the charcoal. Cook the turkey over indirect low heat, with the lid closed, for 1 hour.
After 1 hour, to maintain the heat, add 10 to 12 unlit charcoal briquettes to the lit charcoal, using long-handled tongs to tuck the unlit charcoal between the lit charcoal. Leave the lid off for about 5 minutes to help the new briquettes light. Add the remaining two wood chunks or drain and add the remaining two handfuls of wood chips to the charcoal. 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, wrap them tightly with aluminum foil. Once again, add 10 to 12 unlit charcoal briquettes to the lit charcoal to maintain the heat. 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 160° to 165°F/70º to 74ºC in the thickest part of the thigh (not touching the bone).
Transfer the turkey to a cutting board, loosely cover with foil, and let rest for 20 to 30 minutes before carving (the internal temperature will rise a few degrees during this time). Save the pan contents to make the gravy.
Strain the pan contents through a large sieve into a large fat separator and discard all the solids. Let stand for 5 minutes. Pour off the pan liquid, discarding the fat. Add enough chicken stock to equal 4 cups/960 milliliters of liquid. In a medium, heavy-bottomed saucepan over medium heat melt the butter. Whisk in the flour and cook until the mixture turns the color of peanut butter, about 5 minutes, whisking frequently. Whisk in the reserved pan liquid and the wine. Bring the gravy to a boil, whisking frequently. Reduce the heat to medium-low and simmer for about 5 minutes, whisking occasionally. If the gravy seems too thin, simmer it longer until it is as thick as you like. If it seems too thick, add more stock. Remove from the heat. Add the parsley and season with salt and pepper. Carve the turkey. Serve warm with the gravy.