Soak wood chips in warm water for at least 20 minutes, or chunks at least 1 hour.
Meanwhile, prep turkey: Remove the leg truss from turkey and discard. Remove neck, tail, and giblets from cavity; save for gravy if you like. Rinse bird inside and out and pat dry. Be sure to wash your sink with antibacterial soap afterward.
Put remaining ingredients in a bowl and mash together with a fork.
Set up a charcoal grill for indirect medium-low heat (300° to 325°): Light 50 briquets in a chimney starter. Open grills' bottom vents fully and close lid vents about halfway. When coals are spotted with ash, about 20 minutes, put an aluminum drip pan (at least as large as turkey) in middle of firegrate. Bank coals on either side of pan, piling them highest against walls of grill. The pan's function is to keep coals from tumbling toward the center, under the turkey. Cover grill and let coals burn to medium-low. Check temperature during roasting; if necessary, add 5 more briquets to each mound about every 30 to 45 minutes to keep temperature steady at 300° to 325°.
Meanwhile, spread about 2 tbsp. herb butter throughout the inside of the main cavity. Starting at the breast end, loosen the skin, working all the way down the legs if you can. Spread the rest of the butter under skin and over tops of legs, thighs, and lastly breast. Pat the bird to even out the lumps and evenly distribute butter. Wrap wing tips and drumstick ends with foil so they don't char.
Drain half of wood chips or chunks. Sprinkle evenly over coals. Set cooking grate in place with its hinged flaps aligned over coals to make replenishing them easier. Whenever heat dips below 300° during cooking, add about 5 briquets to each pile of coals.
Oil a shallow but sturdy grilling rack and put turkey on it, breast side up. Set the rack on a rimmed baking pan just big enough to hold it; then set pan with turkey over indirect heat area (aligned with aluminum pan). Insert a cord-style digital thermometer through thickest part of breast until it touches bone. Arrange cord over indirect heat area. Cover grill.
Grill-roast turkey for about 1 hour, tenting it with foil when it begins to get too dark. Drain remaining wood chips or chunks and add to fire as before. Grill-roast turkey until thermometer registers 165°, 2 to 3 hours more. Use an instant-read thermometer to double-check the temperature on the other side of breast, and measure temperature into the thigh joint (where thigh meets body) as well; here, it should be 170°.
Lift baking pan with turkey off grill. Tip turkey by one end of grilling rack so juices run into pan.
Transfer turkey to a serving platter; save pan juices for the gravy, or just strain the juices, spoon off fat, and serve them as is. Let turkey rest in a warm place 20 to 30 minutes. Garnish with extra herb sprigs if you like and bring to the table for carving.
*Use wood chips or chunks to give your turkey a gently smoky flavor. Aidell's favorite woods for turkey are fruit woods like apple, cherry, or peach; find these at home-supply stores.