In a large cast iron skillet, melt the butter over medium heat. Add the mushrooms and onions and cook without stirring for 3 minutes. Stir after 3 minutes and let the other sides caramelize for 3-5 minutes. When the onions and mushrooms are nicely browned, remove them from the skillet.
Meanwhile, in a small bowl add the beef, Worcestershire, black pepper, salt and onion powder. Use your hands to gently knead the meat with the spices. Divide the meat into two portions, and flatten the patties in your hand to make squares that are the same size of the bread.
Sear the patties in the skillet for 5 minutes on the first side. Flip, and then add a slice of cheese on each patty. Cook the patties for another 5 minutes, or to desired level of cooking (medium, medium-rare, or well-done). Use a meat thermometer to ensure food safety.
Remove the patties from the skillet, place on a plate top with the mushrooms and onions, and tent with aluminum foil to keep warm.
Toast the bread slices in the same skillet, about 2 minutes per side.
Serve the patties topped with mushrooms and onions on the toasted bread smeared with spicy brown mustard.