Place a large cast iron skillet over medium-high heat and add the ground beef, allowing it to develop a rich, deep-brown crust before breaking it apart.
Continue browning until about half of the beef achieves a robust sear.
Remove the beef, leaving any rendered fat in the skillet, then melt the butter and incorporate the diced onion, green bell pepper, and minced mushrooms.
Allow the vegetables to brown undisturbed for 1–2 minutes, stir, and repeat for an additional 1–2 minutes until lightly caramelized.
Return the browned beef to the skillet. In a separate small cup, whisk the cornstarch into the beef broth until smooth, then pour this mixture into the pan along with the ketchup, Worcestershire sauce, salt, and black pepper.
Stir thoroughly and cook until most of the liquid reduces, leaving the mixture slightly saucy, about 3–5 minutes.
Remove the skillet from the heat and fold in the chopped provolone, beginning with half and adding gradually until the desired cheesiness is achieved.
Serve immediately on toasted brioche buns.