Preparation

Make the stock

• Position a rack in the center of the oven, and heat the oven to 400°F. Put the wings and any turkey trimmings, if using, on a large rimmed baking sheet with the carrots, celery, onion, and garlic. Roast until the poultry and vegetables begin to brown, about 1 hour. Scrape everything (including any brown bits on the bottom of the pan) from the baking sheet into a large (about 12 quart) stockpot.
• Add the chicken stock and 2 cups water. Bring to a boil over high heat with the lid ajar. Uncover the pot, reduce the heat to medium low, and bring to a simmer. Continue to simmer, stirring occasionally, until the meat begins to fall off the bones and the vegetables are very soft, about 1-1/2 hours. Strain the stock through a large colander over a large bowl, and discard the contents of the colander. Pour the stock through a cheesecloth-lined medium-mesh strainer into a medium pot. Skim and discard any fat from the stock, and then cool and refrigerate until ready to use.

Dry-brine the turkey

• Rub the salt onto the turkey pieces, and put them on a rack set over a large rimmed baking sheet to catch the drippings. Refrigerate, uncovered and skin side up, for at least 3 hours or up to 3 days.

Cook the turkey

• Position a rack in the center of the oven, and heat the oven to 250°F. Put the turkey pieces in a large Dutch oven or roasting pan with sides deep enough to submerge all of the turkey parts (about 16x13x5 inch). Pour the fat over the turkey until completely covered. (If you need a little more, you can top it off with vegetable oil.) Avoid filling to the brim of the pan, which can lead to spillover or serious injury when removing the bird. Place a piece of parchment directly onto the surface of the fat, and then cover tightly with aluminum foil. Roast until the meat is easily pierced with a toothpick, about 3 hours. Carefully remove the pan from the oven, and allow to cool at room temperature for about 1 hour.

Make the gravy

• In a medium saucepan, combine the flour and duck fat. Cook over medium-low heat, stirring constantly with a wooden spoon, until the mixture is thick, smooth, and fragrant, about 2 minutes. While continuously whisking, gradually pour in 3-1/2 cups of the stock, and continue to whisk until the gravy begins to thicken, 12 to 15 minutes. Thin the gravy with any remaining stock (or water) to your desired consistency, if you like. (Refrigerate or freeze the remaining stock for another use.) Season to taste with salt and pepper. Garnish with the parsley, if using.

Sear and serve the turkey

• Working carefully as the turkey will be tender and delicate, remove the pieces from the fat, and transfer to a plate (two fish spatulas work well for this task). Blot with paper towels to remove most of the excess fat, then place skin side down in a cast-iron or other heavy-bottomed pan. Cook over medium heat, searing until the skin becomes crisp and golden, rotating the drumsticks so that they are evenly browned on all sides, about 8 minutes. Serve with the gravy.