Prepare grill for a medium-high indirect cook.
Remove giblets from turkey (discard or save for another use). If you're planning to brine the turkey, omit the salt from the recipe. Blot surface of turkey with paper towel.
Mix together the remaining ingredients in a food processor. Place half of the mixture in a saucepan. Use the remaining half to season the turkey. Apply all over the turkey, making sure to get under the skin of the breast.
When the grill temperature has reached about 350° F, place turkey on the grill rack directly over the drip pan filled with 1 1/2 cups hot water. Grill, covered, over indirect medium heat for 1 hour before basting.
To prepare baste, heat reserved butter mixture on stove top for 3-4 minutes, over medium heat until butter has melted. Remove from heat and let stand for a few minutes before using.
Continue to grill turkey 1 hour-2 hours longer or until a meat thermometer reads 185° F in the thigh. Baste frequently with heated butter sauce every 20 minutes until 10 minutes before removing the turkey from the grill.
Once cooked, carefully remove turkey and place onto a large cutting board. Loosely cover turkey with aluminum foil and let stand for 15 minutes before carving.