Begin by heating a cast iron skillet over medium-high heat. Add the ground beef, allowing it to cook undisturbed for about 5 minutes, or until browned. Stir occasionally to ensure even browning and caramelization. Transfer the meat to a plate and set aside.

In the same skillet, lower the heat to medium and add a tablespoon of butter. Once melted, toss in the sliced onions, bell peppers, and mushrooms. Cook for 3-4 minutes, stirring after the first 2 minutes until tender and fragrant.

Return the browned beef to the skillet, mixing it with the sautéed vegetables. In a small bowl, whisk together beef broth and cornstarch until smooth. Pour this mixture into the skillet along with ketchup, Worcestershire sauce, salt, and pepper. Stir well and bring to a gentle simmer.

Allow the mixture to simmer on medium heat for 3-5 minutes, stirring occasionally until the sauce thickens and coats the beef and vegetables nicely.

Once thickened, turn off the heat and stir in the chopped provolone cheese. Continue stirring gently until the cheese is fully melted and the mixture becomes creamy.

To serve, spoon the hot beef and cheese mixture onto toasted brioche buns. Ensure each bun gets a generous helping for the best flavor.