Gather the ingredients.

Remove the liver from the giblets from a whole turkey or chicken and refrigerate.

Place the remaining giblets into a medium saucepan and cover with 4 cups of cold water. Bring to a boil over high heat. Reduce the heat to low and simmer until the giblets begin to tenderize, about 1 ½ to 2 hours. Add the liver and continue to simmer, 30 minutes more.

Place a mesh strainer or colander over a large bowl. Drain the giblets. Set the liquids aside to use in the gravy, if needed. Let the giblets cool. Remove the meat from the neck and coarsely chop with the rest of the meat.

Over low heat, melt 4 tablespoons (2-ounces) unsalted butter in another medium saucepan. Stir in 4 tablespoons all-purpose flour. Cook, stirring continuously until the roux just barely begins to turn golden, 3 to 5 minutes.

If you don't have drippings from a roasted turkey or chicken, or if you only have a small amount, add the giblet broth in the reserved saucepan, or enough chicken or turkey stock to make 2 cups pan drippings. Slowly stir the drippings and/or broth into the roux. Add ½ cup milk, or half-and-half. Continue to cook, stirring frequently until thickened, 10 to 15 minutes.

Taste and season the gravy with ½ teaspoon fine salt and ½ teaspoon freshly ground black pepper.

Stir in 2 large coarsely chopped hard-boiled eggs and the giblets. Adjust the seasoning, and serve.