Injection brine the turkey: Stir the brine ingredients in a measuring cup until the salt and sugar dissolve. Inject the brine into the turkey, pushing the needle all the way in, then slowly pulling the needle out while depressing the plunger to inject the brine. Try to inject brine into the meat about every inch to inch and a half. First do the breast: working from the front of the bird through the neck cavity, make three to four evenly spaced injections on each side. Next, do the wings – inject a little brine into the knob of the drumette (where the wing meets the breast), and between the two bones in the wing, going in the long way if you can. Move to the back of the bird and inject the thighs through the back cavity – each thigh should take two or three injections. Finally, inject the drumsticks with two injections, one in the knob of meat above the bone, one in the knob of meat below the bone. Mix the soy sauce and paprika, then brush the whole bird with the soy sauce baste. Set the bird on a rack over a roasting pan and refrigerate for at least 8 hours.

Truss and spit the turkey: Two hours before cooking, remove the turkey from the refrigerator. Truss the turkey, following the instructions in my How to Truss and Spit a Turkey post. Skewer the turkey on the rotisserie spit, securing it with the spit forks. Fill a gallon zip-top bag with ice, and lay the bag on top of the turkey’s breast to keep the breast colder than the drumsticks.

Set up the grill for Indirect Medium heat (325°F): Set the grill for indirect medium heat (325°F to 350°F) with the drip pan in the middle of the grill. Set up all the heat in the grill on one side, facing the turkey legs. With a charcoal grill, light a charcoal chimney 3/4 full of charcoal, and wait until it is covered with gray ash. Make a U of charcoal, with the pan in the middle, and the breast facing the open part of the U. With a gas grill, if possible, turn on two burners on one side of the grill instead of one burner on each side. (If you have an infrared rotisserie burner, leave it off at first. Check the turkey after 2 hours of cooking; if the turkey needs last minute browning, turn the infrared burner on high for the last half hour of cooking.)

Rotisserie grill the turkey: Put the spit on the grill, start the motor spinning, and make sure the drip pan is centered beneath the turkey. Close the lid and cook until the turkey reaches 155°F in the thickest part of the breast, about 2 1/2 hours for a 12 pound bird. If you are cooking on a charcoal grill, add 24 coals to the fire every hour to keep the heat going.

Serve: Remove the turkey from the rotisserie spit and remove the twine trussing the turkey. Be very careful – the spit and forks are blazing hot. Let the turkey rest for 15 to 30 minutes, then carve and serve.