- Preheat oven to 400 degrees F (200 degrees C).
- Combine onion, carrots, celery, and vegetable oil in a large roasting pan and toss to coat. Place turkey wings on top of vegetables
- Place roasting pan in the preheated oven and cook until the turkey wings are browned and vegetables are caramelized and softened, 45 to 60 minutes.
- Transfer turkey wings and vegetables to a large stockpot. Place the roasting pan over a stovetop burner on medium heat. Pour 2 tablespoons cold water into the pan and bring to a boil, scraping up any browned bits. Transfer mixture to the stockpot and add 10 cups cold water, thyme, and garlic.
- Bring turkey wing mixture to a boil. Reduce heat to low and simmer, uncovered, until meat falls off the bone, about 3 hours. Skim off turkey fat throughout the process and set aside 2 tablespoons.
- Strain turkey stock and reserve 6 cups of stock; discard all the solids.
- Heat butter and 2 tablespoons reserved turkey fat in a large saucepan over medium heat. Sprinkle in flour and cook, whisking continuously, until it begins to smell like cooked pie crust, 2 to 3 minutes. Slowly pour in turkey stock, whisking continuously. Increase heat to high and simmer until thick and warmed through, about 5 minutes. Season with salt, black pepper, and cayenne pepper to taste.