Place turkey giblets and neck into a saucepan with 4 cups water and bring to a boil; reduce heat to low. Simmer until broth is reduced to 3 cups, about 1 hour. Strain broth and reserve 1/2 cup of giblets; chop.
Combine giblet broth and turkey drippings in a saucepan over medium heat. Whisk cornstarch in 1/2 cup cold water in a bowl until smooth. Whisk cornstarch into stock and drippings. Bring to a boil, reduce heat to low, and stir chopped giblets and hard-cooked egg into the gravy. Season with salt and black pepper. Simmer until gravy is thickened, about 5 minutes.