Carefully drain excess fat from the skillet, leaving any browned bits. Return the pan to medium-low heat and add the butter. When it melts and begins to foam, add the sliced onion and cook gently 4–5 minutes until softened and translucent.
Stir in the sliced mushrooms and sauté another 3–4 minutes until they begin to release their juices.
Sprinkle the flour over the onion and mushroom mixture, stir to combine, and cook 1–2 minutes to remove the raw flour taste.
Increase heat to medium and slowly add the beef broth while scraping up any browned bits from the bottom of the skillet. Once the sauce comes to a simmer, nestle the patties back into the pan.
Lower the heat to maintain a gentle simmer, cover, and cook about 20 minutes. Flip the patties 2–3 times during cooking. Uncover for the final 5–8 minutes to allow the gravy to thicken, adjusting heat as needed to keep a steady simmer.
Taste the gravy and season with additional salt and pepper if necessary.