Pour the turkey drippings into a large measuring cup. Let it rest for about 10 minutes, then spoon off the fat and discard. If you have one of those liquid measures made for this purpose, then this process will be even easier.
Add enough chicken broth to the drippings to total about 4 cups of liquid in all.
Set the roasting pan, lovely brown bits included, over top burners that are set on medium heat. Add butter and melt, scraping up the browned turkey bits with a wooden spoon. Add flour and, using a whisk, stir until this mixture (called a roux) reaches a dark brown color, 4 to 5 minutes.
Whisking constantly, slowly add the broth mixture to the pan. Bring the mixture to a boil, whisking constantly. Reduce the heat slightly so that the mixture is simmering. Cook, whisking frequently, until the gravy is thick and smooth, about 15 minutes. Lower heat and keep warm, whisking occasionally to stop a skin from forming, until ready to serve.