Tip:If you make this gravy in a saucepan instead of the roasting pan, make sure you get all the drippings from the roasting pan by pouring some of the giblet broth into the hot roasting pan after you've poured off the liquid drippings from the roasting pan. Scrape with a wooden spoon to loosen any cooked-on drippings, and then use this mixture as part of your gravy liquid.
Heat the giblet broth. Pour off all the juices and drippings from the roasting pan into a 1-quart heatproof measuring cup. Let the fat rise to the top and then spoon 1/4 cup of the fat back into the roasting pan. Skim off and discard as much of the remaining fat as possible from the juices. Add the giblet broth to the skimmed juices. If necessary, add chicken broth until you have a total of 4 cups of liquid.
Set the roasting pan over two burners on medium heat. Sprinkle the flour into the pan. Stir with a flat whisk or wooden spoon and cook for about 2 minutes.
To keep lumps from forming in the gravy, slowly pour about 1/2 cup of the broth mixture into the pan while whisking vigorously to disperse the flour evenly into the liquid. The liquid should thicken quickly and look almost gluey. As soon as it thickens, add another 1/2 cup or so of broth while whisking. Repeat until the gravy starts looking more like a smooth sauce than glue.
At this point, it's safe to whisk in the remaining broth and bring the gravy to a simmer. Add the thyme sprigs and simmer for about 5 minutes. Strain the gravy through a medium mesh sieve, season with salt and pepper, and serve in a heated gravy boat or other vessel.