Using a large skillet, brown the ground beef over medium heat. Drain and set aside.
In same skillet, melt butter. Add onion, peppers, mushrooms. Cook until soft.
Stir in cooked ground beef.
Mix in ketchup, Worcestershire sauce, salt, and pepper.
In bowl, mix cold beef broth and cornstarch. When combined, pour into skillet and stir.
Continue to cook, stirring constantly until sauce is thick
Mix in cheese. Stir until melted and combined.
To Serve: toast buns and spoon sloppy joe mixture onto bottom half of the bun.