Brine the turkey: Bring 2 cups water, the salt, brown sugar, 1 halved garlic head and 10 thyme sprigs to a boil in a medium saucepan over medium-high heat, stirring to dissolve the salt and sugar. Remove from the heat and add 6 cups cold water; let cool to room temperature. Whisk the buttermilk, hot sauce, mustard and sliced onion in a large bowl, then whisk in the salt-sugar mixture. Place the turkey in a brining bag or extra-large pot. Pour the brine over the turkey, seal or cover and refrigerate overnight.
Position a rack in the lowest part of the oven; remove the other racks and preheat to 350˚. Set a rack in a large roasting pan. Remove the turkey from the brine, letting the excess drip off as much as possible, and transfer, breast-side up, to the rack. Pat the turkey dry. Let sit at room temperature 30 minutes.
Combine the paprika, onion powder, granulated garlic, cayenne and 1 teaspoon black pepper in a small bowl. Stuff the turkey cavity with the remaining 1 halved garlic head and 10 thyme sprigs and the quartered onion; tie the legs together with kitchen twine. Tuck the wings under the body. Season the turkey all over with salt and brush with the melted butter. Coat the turkey with the spice mixture.
Roast the turkey 1 1/2 hours, then baste with the pan drippings. (Add 1/4 cup water to the roasting pan if it’s too dry.) Continue to roast, basting every 30 minutes, until the skin is crisp and a thermometer inserted into the thigh registers 165˚, 45 minutes to 1 1/2 hours. Transfer the turkey to a cutting board and let rest 30 minutes before carving.
If you’re planning to brine, don’t buy a kosher or self-basting turkey (it’ll turn out too salty), and add the drippings to your gravy gradually so you can control the saltiness.