Remove neck and giblets and discard excess fat. Rinse the bird inside and out and pat dry. Season body cavity with salt and pepper. Tie legs together and twist wing tips under back. Brush turkey with oil and lightly sprinkle with salt and pepper. Arrange turkey breast side up, in center of cooking grate. Place lid on grill.
Cook 11 to 13 minutes per pound or until an instant read thermometer inserted into the thickest part of the thigh not touching bone registers 180 degrees and the juices run clear. Transfer turkey to a platter. Let stand for 15 minutes before carving.
Ignite the grill and turn all burners to high. Close the lid and preheat 10 to 15 minutes to bring the grill to 500 to 550 degrees. Turn burners down to medium heat. Place turkey breast side up in a roast holder inside a foil drip pan. Position pan in the center of the cooking grate and place the lid on the grill. For three burner grills, set front and back burners to medium and turn off center burner. For two burner grills, turn the front and back burners to medium. Keep grill covered.