Bring giblets, turkey neck, stock, and water to a boil. Reduce heat and simmer for 2 hours.
Place a strainer over a bowl and drain the giblets. Transfer giblets to a cutting board and allow to cool. Remove the meat from the neck and chop all the meat from neck and the rest of giblets.
Melt butter in a saucepan and stir in flour. Whisk continuously for 4-5 minutes. Slowly pour and whisk drippings from the turkey and 2 cups of remaining broth into the butter-flour mixture in the saucepan and boil for 3 minutes. Add the milk and continue stirring until mixture is thickened. Stir in chopped boiled eggs.
Season with kosher salt and freshly ground pepper to taste.
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