Begin by placing the ground beef in a large cast iron skillet to enhance browning. Cook until a deep brown crust forms before crumbling the meat. Continue to cook until about half of the beef develops a similar crusty exterior.

After removing the beef, maintain the drippings in the pan and melt in the butter. Add the onions, bell peppers, and mushrooms to the skillet. Allow them to brown for 1-2 minutes before stirring. Repeat the process to ensure the vegetables develop a rich, caramelized color.

Reintroduce the beef to the skillet. In a separate container, blend the beef broth with the cornstarch until smooth. Incorporate this mixture into the skillet along with ketchup, Worcestershire sauce, salt, and black pepper. Stir and cook until the sauce thickens and most of the liquid is absorbed, which should take about 3-5 minutes.

Remove the skillet from the heat and mix in the chopped Provolone cheese until it melts and blends into the meaty mixture seamlessly.

Serve the savory filling hot, spooned generously onto the toasted brioche buns.