Make the brine. In a large saucepan, combine 4 cups water, kosher salt, maple syrup, garlic, rosemary, thyme, lemon zest, bay leaves, and peppercorns. Heat over medium-high heat, stirring occasionally, until the salt dissolves and the brine is just about to boil. Remove from the heat.
Cool the brine. Pour the 4 cans of cold beer and/or hard cider into the saucepan to help cool the brine down quickly.
Prepare the brining bag. Add 8 cups of ice to a large brining bag. Pour in the brine, then add 4 cups of cold water and stir. Make sure the brine is completely cold (about 40°F) before adding the turkey.
Brine the turkey. Place the turkey breast side down in the brine, working some of the liquid into the cavity. Add more cold water if needed so the turkey is mostly covered, or prop one side up so the liquid covers the bird. Tie the bag closed, removing most of the air from the bag, and refrigerate for 24 hours.
Air-dry the turkey. Remove the turkey from the brine and discard the brine. Place the turkey on a wire cooling rack set on top of a sheet tray. Refrigerate, uncovered, for another 24 hours to dry out the skin. Do not add any more salt.
Preheat the grill. On smoking day, fill your pellet grill hopper and preheat to 300°F. Place a water pan in the hottest area of the grill.
Make the maple herb butter. In a small saucepan, melt the butter with 2 tablespoons maple syrup, rosemary, and thyme over low heat. Stir, then remove from the heat and let cool for a few minutes, until it starts to thicken and is spreadable but not runny.
Butter and season the turkey. Pat the turkey completely dry with paper towels. Slather the maple herb butter all over the outside of the skin, then season with coarse black pepper.
Smoke the turkey. Set the turkey on the grill grates. Smoke at 300°F until the thickest part of the breast reaches about 130°F.
Crisp the skin. Increase the grill temperature to 350°F. Continue cooking until the thickest part of the breast reaches 155°F and the thigh reaches at least 160°F. *See notes.
Rest the turkey. Remove the turkey from the grill and let it rest for 30-45 minutes, leaving the thermometer probes in place. Carryover cooking will bring the breast up to 165°F. *See notes.
Carve and serve. Carve the turkey and serve.