Combine giblets, neck, and water in a saucepan. Bring to a boil; cover, reduce heat, and simmer 1 hour or until giblets are tender. Strain, reserving broth. Discard turkey neck. Coarsely chop giblets; set aside.
Skim and discard fat from reserved pan drippings of roasted turkey. Add reserved broth (at least 2 1/2 cups) to pan drippings; stir until browned bits are loosened from bottom of roasting pan.
Transfer broth and drippings to a saucepan, if desired, or continue cooking in roasting pan over 2 burners on the stove. Stir in chopped giblets. Add onion and next 3 ingredients. Bring to a boil; reduce heat, and simmer, uncovered, 30 minutes.
Combine cornstarch and 1/4 cup water, stirring well; gradually stir into gravy. Bring to a boil; boil 1 minute or until thickened. Stir in egg. Add salt and pepper to taste. Serve hot.