Begin by browning the lean ground beef in a hot pan. Once the beef is browned, remove it from the pan and set it aside.
In the same pan, melt the butter and sauté the diced onions, bell peppers, and minced mushrooms until they are nicely browned.
In a bowl, combine the beef broth, cornstarch, ketchup, salt, and pepper; pour this mixture into the sautéed vegetables and let it simmer until it thickens.
Return the browned beef to the pan and mix everything together well.
Turn off the heat and stir in the provolone cheese until it is completely melted.
Butter the brioche buns and toast them separately in a pan until they turn golden brown.
Fill each toasted bun with the cheesy beef mixture and serve warm.
