In a small saucepot, combine salt, sugar, and 4 cups of water and bring to a boil. Once sugar and salt dissolve, remove from heat and let cool to room temperature. Stir in buttermilk. Transfer brine to a large container and submerge turkey. (If the bird isn’t completely submerged, turn it breast-side down so the backbone is out of the brine.) Set aside in refrigerator for 24 hours.
Remove turkey from brine, letting excess drip off, and transfer to a rimmed baking sheet. Set aside, uncovered, in fridge for 2 days. (This helps dry the skin, so it can crisp up more in the oven.)
Preheat oven to 325 degrees. Add carrots, celery, and onion to a large roasting pan and top with peppercorns, thyme, bay leaves, and garlic. Set turkey on top and roast until breast and thigh reach an internal temperature of 165 degrees, about 2½ hours. Remove from oven and let rest for 20 to 30 minutes before carving.