Heat the butter in a large cast iron skillet over medium high heat. Add the sliced onions and thyme. Cook for 2 minutes, or until onions are softened. Add the mushrooms and saute for another 4 minutes or until mushrooms are golden brown. Pour in the broth and simmer for 3 minutes or until broth as cooked down and most of the liquid has evaporated. Season with 1/4 teaspoon salt.

Remove onion mixture from skillet and set aside. Leave drippings in the pan.

Heat the skillet with the drippings to medium high heat (add more butter or duck fat to the skillet if needed). Season the filet mignon on all sides with salt. Place the filets in the skillet and cook for 5 minutes per side for medium rare steak (or longer if preferred).

Remove steak from the skillet and allow to rest for 5 mintutes.

Divide the onion mixture onto 4 serving plates. Top the onion mixture with a filet mignon. Serve with bearnaise sauce if desired.