In a large skillet over medium heat, add ground beef and cook while breaking apart until no longer pink. Drain grease.
Return beef to the skillet then add onions, green peppers, and mushrooms. Cook for an additional 5 to 7 minutes, or until onions are tender.
In a small separate bowl, combine beef broth and cornstarch. Whisk until no lumps remain. Pour broth over meat and stir well.
Add ketchup, Worcestershire sauce, A1 sauce, salt, and pepper. Stir well. Let cook until most of the liquid has been absorbed, about 5 minutes.
Turn off the heat to the burner. Add sliced provolone cheese to the top of the beef and allow to melt.
Serve meat and cheese mixture on sesame seed hamburger buns.
