Light a Big Green Egg or large grill to a medium flame. Once lit, regulate temperature with the bottom and top vents to maintain a temperature of about 200 to 225 degrees. (Carter prefers fruitwoods like cherry for smoking turkeys.)
Roughly chop onion, celery, and carrots. Place in a large bowl and add the sage, thyme, and rosemary sprigs. Add butter and mix to combine. Season inside and outside of turkey with salt and pepper. Fill the turkey’s cavity with the vegetable-and-herb butter.
Place turkey on the grates of the grill and close the lid. Maintain a temperature of 200 to 225 degrees. Smoke the turkey for approximately 6 hours. Rotate occasionally. When internal temperature reaches 140 degrees, remove turkey. Let rest for 15 minutes. Carve and serve.