Preparation

To brine the turkey:

• Combine all the brine ingredients in an enamel or stainless-steel pot big enough to hold the brine and turkey. Bring to a simmer, remove from the heat, and let cool completely. Remove the neck and giblets, rinse the turkey well, and put it in the cold brine; add water if the brine doesn’t cover the bird. Refrigerate for 2 to 4 days, turning the bird twice a day.

To cook the turkey:

• Remove the bird from the brine, pat it dry, lightly brush it with olive oil, and set aside. Prepare the grill by lighting about 30 charcoal briquettes or small pieces of hardwood charcoal, preferably in a chimney starter. When the coals are hot and spotted gray, put an aluminum-foil drip pan that’s at least 1 inch deep in the middle of the grill. Arrange half the coals on one side of the pan and half on the other. Put about 1/2 cup of wood chips in a double layer of aluminum foil and set them on the hot coals.
• Put the upper rack of the grill in place and center the turkey, breast side up, on the rack over the drip pan. Cover the grill and partially close the air vents. Regulate the vents to keep the wood chips smoking and the coals burning slowly, checking every 25 minutes or so. Add charcoal periodically. Keep the temperature in the grill between 275° and 325°F.
• Add more wood chips as you need them. Keep the smoke going for 1-1/2 to 2 hours; then remove the chips and continue cooking without smoke until the bird is done. The total cooking time for a 12- to 14-lb. bird is about 3 to 3-1/2 hours. Test the turkey with an instant-read thermometer in the thickest part of the thigh or breast. You can also cut a small incision at the leg-thigh joint to see that the juices run clear. When the internal temperature reaches 165°F, remove the turkey from the grill. Let it rest at least 20 minutes before carving.