Place the giblets in a saucepan, cover with water and bring to a boil. Cook until the giblets are cooked through, 20 to 25 minutes. Remove the pan from the heat and let the giblets sit in the water, allowing them to cool.
Remove the giblets from the saucepan and chop the liver and gizzards. Pick the meat from the neck bone and discard the bones. Save the giblet broth.
Pour the turkey drippings from the roasting pan into a pitcher. Allow the fat to rise to the top, leaving the stock juices at the bottom. Ladle or pour the fat into a separate container.
Place the roasting pan on the stove over medium heat. Add 1/4 to 1/2 cup of the reserved fat. Sprinkle over the flour and whisk to combine, scraping the brown bits from the bottom of the pan. If it is too oily, sprinkle in another 1 to 2 tablespoons of flour. If it is too dry, add more fat and stir until the flour and fat are combined. Continue to cook the roux until it is golden brown, 4 to 5 minutes.
While whisking, pour in the chicken stock. Cook, whisking gently, until the mixture begins to thicken. Pour in the stock juice drippings from the pitcher and continue to cook, stirring, until thick, 4 to 5 minutes. Thin as needed with the giblet stock and additional broth if needed. Stir in the giblets and season to taste with salt and black pepper. Cook for another 5 minutes or so until the giblets are warmed through, then serve.