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 grill for indirect medium-low heat (300° to 325°): Set an aluminum drip pan (at least as large as turkey) under the cooking grate and on middle burner (for a 3-burner grill) or on one of the two burners (for a 2-burner grill). With grill lid open, ignite all burners and turn to high. Close lid and heat 10 minutes; turn off burner under the drip pan and adjust other burners to maintain 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 down the legs if you can. Spread the rest of the butter under the skin and over top 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. Put chips in a smoker box, if your grill has one, or put chips in a small shallow disposable aluminum pan and set over direct heat. If using chunks, set directly on grate over a lit burner.
Oil a shallow but sturdy grilling rack and put turkey on it, breast side up. Set the rack directly on cooking grate over the drip pan. Insert a cord-style digital grilling 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 if 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°, 1 to 2 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 the turkey onto 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.