Using a fine mesh sieve, strain turkey pan drippings into a large measuring cup. Skim off the fat, reserving 1/3 cup in a small bowl. Place the roasting pan over medium heat, add 1 1/2 cups low-sodium chicken broth and 1/2 cup beer and bring to a boil. Stir, scraping up any browned bits, until the broth reduces slightly, about 2 minutes.
Add enough remaining broth to the cup with the pan drippings to measure 1 1/2 cups, pour it into the roasting pan and bring to a boil. Whisk 1/4 cup plus 2 tbsp. flour into the reserved 1/3 cup fat, then whisk the flour mixture into the pan juices; return to a boil. Simmer, whisking, until thickened, about 5 minutes. Season with salt and pepper.