Return the skillet to medium heat, add 1 tablespoon butter and the onion and cook, stirring frequently, until softened and golden, about 10 minutes; season with salt and pepper. Push the onion to one side of the skillet and melt the remaining 1 tablespoo butter on the other side. Whisk the flour into the butter and cook, whisking, for 2 minutes. Whisk the beef broth into the flour mixture, incorporating the onions, and bring to a boil, whisking frequently, until the gravy has thickened. Cover and remove from the heat.
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