Strain drippings and vegetables into a bowl (do not clean pan) and press to extract as much liquid as possible; discard vegetables. Skim fat from juices and reserve 1/2 cup.
Set roasting pan across 2 burners over medium heat and add fat. Slowly stir in flour until smooth and cook, stirring, until roux is a rich golden brown, 3 to 4 minutes.
Gradually whisk in strained juices, stirring and scraping up any browned bits. Add broth, bring to simmering over medium-high heat, and simmer until thickened, about 5 minutes. Add chopped neck and giblet meat and pour into gravy boats. Float egg slices on top if you like.
*Follow recipe for Easy Chicken Broth at sunset.com, using chicken or turkey bones.