In a 5-gallon food-grade bucket, combine the water, salt, brown sugar, peppercorns and allspice.

Remove the neck and giblets from the turkey and discard or set aside for later use. Remove any plastic or wire holding the turkey legs together and discard. Rinse the turkey well, inside and out, then gently lower it into the bucket of brine, making sure it is fully immersed. Place the bag of ice on top of the turkey to keep it submerged fully in the bucket. Cover the bucket a lid or a thick, tight layer of aluminum foil and set aside to brine overnight.

When ready to smoke the turkey, remove the bird from the brine, drain well and pat the skin and inside cavities dry with paper towels. Fold the wingtips behind the back and tie the legs together with butcher’s twine. Rub the turkey inside and out with the oil, then stuff both the main cavity and the neck cavity with the quartered apple, onion and fresh herbs.

Start an electric smoker on the Smoke setting with the lid open until the fire is established (4 to 5 minutes). Place the turkey directly on the grill grate, breast side up. Close lid and let the turkey smoke on the Smoke setting (between 150 and 180 degrees) for 1 ½ to 2 hours.

Combine the melted butter and rosemary in a small bowl. After the initial Smoke, leave the turkey on the smoker and brush all over with the rosemary butter, then turn the smoker heat up to 350 degrees. Smoke the turkey for an additional 2 to 2 ½ hours, or until the internal temperature of the turkey is 170 degrees Fahrenheit on an instant-red meat thermometer stuck in the thickest part of the thigh.

Carefully remove the turkey from the smoker and let it rest for 20 minutes before carving.