Heat a cast iron skillet over medium-high heat. Add ground beef and cook until it's half browned, about 4–5 minutes, then stir to develop a flavorful crust. Once fully browned, about 6–7 additional minutes, remove the beef from the skillet and set it aside.

In the same skillet, melt 2 tablespoons of butter over medium heat. Add diced onions, bell peppers, and mushrooms, then let them sauté undisturbed for 3–4 minutes. Stir occasionally until they soften further, about an additional 2–3 minutes.

Return the cooked ground beef to the skillet with the sautéed vegetables. Whisk together 1 cup of beef broth and 1 tablespoon of cornstarch, then pour this mixture into the skillet along with ¼ cup of ketchup, 1 tablespoon of Worcestershire sauce, and season with salt and pepper. Stir everything together.

Lower the heat and let the mixture simmer for 3–5 minutes until thickened. Stir occasionally.

Stir in chopped provolone cheese and mix until melted into the filling.

Toast brioche buns lightly, then spoon cheesy beef filling onto each bun. Serve immediately.