Melt the butter in a large skillet over medium heat. Add the chopped onion and green bell pepper, and sauté for about 5 minutes until they are soft. Add garlic and cook 1 more minute.
Add the ground beef to the skillet. Break it up with a spoon and cook until it is fully browned, approximately 6 minutes.
Stir in the beef broth, ketchup, Worcestershire sauce, garlic powder, salt, and pepper. Allow the mixture to simmer for 5-10 minutes until it thickens slightly.
Lightly toast the brioche buns. Spoon the beef mixture onto each bun and add a cheese slice. Melt the cheese under a broiler for about 1 minute and serve warm.