Pour pan drippings into a glass measuring cup and let sit until fat rises to the top, about 15 minutes. Skim off most of the fat, reserving ¼ cup; discard the rest. Reserve defatted drippings.
Set roasting pan on top of stove so that it spans two burners; turn burners to medium-low. Spoon ¼ cup turkey fat back into pan (add butter if you don’t have enough). Whisk ¼ cup flour and sage into fat. Cook, whisking, until flour is nicely browned. Slowly whisk in hot turkey or chicken broth until smooth. Whisk in defatted pan drippings and cider until mostly smooth; then use a wooden spoon to scrape up any browned bits from the bottom of the pan. Whisk again until smooth
Increase heat to high and let gravy boil until thick, 3 to 5 minutes. If gravy still seems thin, sprinkle in an additional tablespoon or two of flour, whisking as you go. Season to taste with salt and pepper. Pour gravy through a strainer into a serving boat and serve hot.