Light the grill and set it up for two-zone cooking, establishing the temperature at 350°F. For added flavor, try adding a pecan or oak wood chunk to the hot coals.
While your grill is preheating, remove the backbone from the turkey with a sharp knife or kitchen shears.
Using the knife or shears, cut down the middle of the breastbone to create two halves of the turkey. Rinse each half under cold water, and pat dry with paper towels. Then rub each half with olive oil and season the bird liberally with the classic roasted turkey rub.
Place the turkey halves on your grill turkey breast side up, over indirect heat and above a drip pan. Close the lid and allow the turkey to cook for 30 minutes before basting with the melted butter and olive oil. Baste the turkey every thirty minutes until the internal temperature reaches 150°F in the breast or 170°F in the thigh as indicated by a reliable thermometer. (About 2 hours)
Remove the turkey from the grill, cover with foil and allow it to rest for 10 - 15 minutes before carving. Carry-over cooking will raise the internal temperature to USDA standards.