Add the butter to a medium size skillet over medium heat. Once the butter is melted, add the patties to the skillet. Cook for 5 minutes on each side, until cooked through. Remove the patties from the skillet and set on a plate. Cover the plate with foil or a lid to keep warm.
Add the butter in the skillet and use a spoon to scrape up all the bits from the bottom. Add the onions and cook for 5 minutes until translucent.
While stirring, add the broth, stirring until all the flour is mixed in. Cook for 3 to 4 minutes, stirring often, until it begins to thicken. Season with salt and pepper, tasting as you season.
Add the mushrooms and cooked patties back to the skillet and reduce the heat to low. Simmer for 7 to 10 minutes, until the mushrooms are tender and the gravy is thick.