Bring stock, water, and turkey neck and giblets to a boil in a saucepan. Reduce heat, and gently simmer, uncovered, for 30 minutes.
Meanwhile, pour pan drippings into a clear measuring cup or a gravy separator, and let stand until separated, about 10 minutes. Pour off fat.
Pour broth and giblets through a fine sieve, discarding solids. Return broth to pan, add defatted drippings, and bring to a boil over medium heat.
Place flour in a heatproof bowl. Whisking constantly, pour in 1/2 cup boiling-hot broth mixture until it forms a paste. Whisk paste into remaining broth in pan, and boil for 3 minutes. Stir in eggs if desired, and season with salt and pepper.
Gravy can be refrigerated airtight for up to 3 days; reheat before using.