Melt butter in a skillet over medium-high heat.
Add the onion, bell pepper, and mushrooms and cook until tender, 4-5 minutes.
Add the ground beef and cook until no longer pink; drain off excess grease and return the mixture to the skillet.
In a small bowl, whisk together beef broth, ketchup, and cornstarch.
Mix the broth into the beef mixture and cook over medium-high heat 4-5 minutes, until the mixture has thickened.
To serve, spoon a generous portion of the beef mixture onto a hamburger bun and top with shredded cheese.
Enjoy!