Remove turkey from the roasting pan. In a small jar, shake flour with water until smooth; set aside.

Leaving carrot, onion, and turkey neck in the pan, remove 2 to 3 tablespoons of fat from the bottom, using a baster.

Place pan over medium heat; add chicken broth. Scrape the bottom of the pan with a wooden spoon to loosen any browned bits. Simmer the mixture until the sauce has darkened, about 5 minutes.

Remove the vegetables and the turkey neck from the pan, and discard. Strain the sauce into a large measuring cup; discard any solids. You should have approximately 3 1/2 cups of strained liquid; add more stock if needed.

Return the liquid to the pan; bring it to a simmer. Whisking constantly, gradually add the flour mixture.

Cook mixture until gravy is smooth, has no lumps, and is thick enough to coat the back of the spoon, 10 to 12 minutes. Season gravy with salt and pepper to taste. Serve hot.