Our family has been cooking Thanksgiving turkeys n a weber grill since the mid 1980's. Turkey cooked on weber grill is moist, with a smoky flavor and dark crispy skin. In this video taken in 2006, Mark demonstrates the technique:
Start with 50 briquettes and add 16 (8 per side) every hour.
small stuffed turkey -- 3 hour total
large stuffed turkey -- 4 hour total