Boil broccoli in 1-inch of water, covered, 3 minutes. Remove from heat. Drain, but leave covered in pan for another 5 minutes. Broccoli will continue to cook, but not lose all of its color.
In a small skillet, heat butter and oil over medium high heat. Cook mushroom caps 3 or 4 minutes until just golden, turn them and set to sides of pan. Add the meat seasoned with salt and pepper and cook 3 minutes on each side for medium rare, 5 minutes on each side for medium well. Remove steaks and caps from the pan and cover with loose foil tent to keep warm and let juices distribute. Add butter and flour to the pan and whisk in wine. Scrape up drippings, reduce wine for half a minute, then whisk in a little beef stock. Thicken sauce another half minute, then remove from heat.
To serve, top each steak with a mushroom cap and pour a little sauce down over the top. Serve with the broccoli florets alongside.