Mix brine ingredients into water and stir until salt is completely dissolved. Place rinsed turkey into brine mixture in a food-safe container (roasting bags are good) with enough brine mixture to cover completely when bag is cinched. Refrigerate bird in brine mixture for about 12 hours.
Drain bird and pat dry. Meanwhile, soak mesquite chips in enough water to cover for approximately 30 minutes. Drain chips and place in small handfuls onto foil and make pouches with little holes pricked in the foil with a fork.
Prepare grill for indirect heat using a drip pan. Place liquid of your choice into drip pan (water or low carb beer is good). Rub turkey breast with softened butter on the outside and season with salt and pepper (add garlic if you like). Oil grill screen and put a pouch of mesquite chips directly on the coals.
Place turkey over the drip pan and cover grill. Grill for several hours, turning turkey as needed, and adding a pouch of chips and a few new coals about every 30-45 minutes to maintain a steady heat. Cook until meat in the deepest part of the breast reads 170°F (you will need an instant read thermometer for this). Allow bird to sit for 15-30 minutes, undisturbed, before carving to maintain juiciness.