In a large stockpot, combine 1 quart apple cider, the salt, allspice, and cloves and bring to a simmer over medium high heat. Stir until the salt is dissolved.
Remove from the heat and stir in the remaining 2 quarts apple cider, and the water. Let the brine cool to room temperature. If necessary, transfer the brine to a large stock pot, canning pot, or other vessel that will accommodate the turkey snugly.
Add the turkey to the brine, ensuring that it's submerged. Cover and refrigerate for 24 hours.
Drain the turkey and pat it dry with paper towel.
Place the turkey breast side down on a work surface and use a pair of heavy-duty kitchen shears or a sharp knife to cut down each side of the back bone. Remove the back bone. The turkey should sort of flop open like a book.
Flip the turkey over so that it's breast side up and the legs are sticking out each side. Push down on the center of the breast bone until you hear a crack.
Place the chicken on a wire rack set in a rimmed baking sheet and refrigerate, uncovered, for 2 hours.
Make the herb butter
Prepare a grill for medium (375°F) indirect heat. Ideally you want to have a burner on either side of the cool zone. (If your grill won't accommodate this, heat one side and leave the other side as the cool zone.)
Mash the butter and herbs together in a small bowl. Reserve half of the herb butter for basting.
Remove the turkey from the fridge and use your fingers to separate the skin from the meat without tearing the skin. Rub the remaining half of the herb butter under the skin, distributing it as evenly as possible.
Place the turkey, skin side up on the cool zone of the grill, cover, and cook for 1 hour. (If you only have burners lit on one side of the grill, I recommend turning the turkey after 30 minutes.)
Melt the remaining herb butter.
After the turkey has cooked for 1 hour, begin basting it with the herb butter every 10 minutes or so, until the turkey is golden brown and has reached an internal temperature of 165°F, 30 to 50 minutes more (about 8 minutes per pound).
Remove the turkey from the grill and let it rest at least 30 minutes before carving.
Notes
Best container--For this size of turkey, I find that a canning pot or large stockpot works well for containing the turkey and brine. If you are using a larger container, you may need to increase the amount of brine in order to submerge the turkey.
Submerge it!--If your turkey isn't submerged in the brine, you can use a plate or a zip-top bag filled with water to help submerge it.
Cover it--If your turkey begins to brown too much during cooking, tent it loosely with foil.
Dietary--This recipe is suitable for gluten-free and dairy-free diets.