Cut the legs and wings off of the turkey; separate the drumsticks and thighs. Using poultry shears, cut off the backbone. (Reserve the wings, neck and backbone for making stock.) Set the breast, thighs and drumsticks on a rimmed baking sheet and sprinkle all over with the 1 1/2 tablespoons of salt. Cover and refrigerate for at least 8 hours or up to 24 hours.
Rinse the turkey parts and pat dry. In a large enameled cast-iron casserole, heat the olive oil. Add the thighs and drumsticks and cook over moderate heat until well browned all over, about 15 minutes. Transfer the turkey to a platter.
Add the garlic, celery, carrot and onion to the casserole. Cover and cook, stirring occasionally, until the vegetables just begin to soften, about 5 minutes. Stir in the tomato paste and cook for 1 minute. Add the wine and cook until reduced by half, about 5 minutes. Return the thighs and drumsticks to the casserole and add 4 cups of the Turkey Stock. Cover and braise over low heat until the turkey is tender, about 1 hour and 15 minutes. Transfer the turkey to the platter; strain the braising liquid and return it to the casserole.
Meanwhile, preheat the oven to 350°. Spread half of the butter under the turkey breast skin; tuck in the sage and thyme. Spread the remaining butter over the skin.
Heat a large ovenproof skillet. Add the turkey breast, skin side down, and cook over moderately high heat until browned, 12 minutes. Turn the breast skin side up and transfer the skillet to the oven. Roast the turkey breast for 1 hour and 15 minutes, until an instant-read thermometer inserted in the thickest part of the meat registers 150°. Transfer the breast to a cutting board and let rest for 15 minutes. Add the remaining 2 cups of Turkey Stock to the skillet and bring to a boil, stirring to scrape up any browned bits on the bottom.
Boil the braising liquid in the casserole until reduced to 3 cups, about 10 minutes. Add the stock from the skillet. In a small bowl, whisk the flour with 1/4 cup of water; whisk this mixture into the liquid in the casserole and boil, whisking frequently, until the gravy is lightly thickened, about 5 minutes. Season with salt and pepper. Transfer the gravy to a gravy boat.
Remove the bones from the thighs and slice the meat 1/2 inch thick. Transfer the thigh meat to a roasting pan along with the drumsticks. Cover with foil and reheat. Carve the turkey breast and arrange on a platter. Arrange the dark meat alongside; serve with the gravy.