Pour the contents of the giblet bags into a large saucepan. Cover with water and boil until cooked through, 20 to 25 minutes. Turn off the heat and set aside. Remove the giblets from the pan, allow them to cool, and then chop up the gizzards and liver. Pick the meat from the neck. Discard the bones.
Pour all the turkey drippings out of the roasting pan into a separate container. Allow grease to separate from the juices. Ladle out the fat into another bowl.
Place the roasting pan over 2 burners over medium heat (there should be lots of yummy brown bits in the pan). Add 3 to 4 tablespoons of the fat back into the pan. Whisk it around to evenly distribute it throughout the pan. Sprinkle 5 to 6 tablespoons of flour over the grease and whisk to combine it with the fat, loosening the bits from the bottom of the pan a little in the process. Cook the roux until it becomes a deep golden brown, 4 to 5 minutes.
Pour in the chicken broth, whisking constantly. Add 2 to 4 cups, depending on your need. Cook the gravy, whisking gently, until the mixture begins to thicken. Pour in a little bit of the turkey drippings, but be aware that because the brine is so salty, these drippings are very salty. Keep cooking the gravy until thick, adding in chopped giblets, to taste, as well as plenty of black pepper. Use the giblet water from the saucepan to thin the gravy if it gets too thick.