Cook the Ground Beef : Heat a large skillet over medium-high heat and add the ground beef. Cook until browned, breaking it apart with a wooden spoon. Drain excess fat if necessary and set aside.

Sauté the Vegetables : In the same skillet, melt the butter over medium heat. Add the diced onion, bell pepper, and minced mushrooms. Cook until the vegetables are soft and fragrant, about 5-7 minutes.

Combine the Beef and Vegetables : Return the cooked ground beef to the skillet with the sautéed vegetables. Stir in the ketchup, Worcestershire sauce, salt, and pepper. Mix well.

Thicken the Sauce : In a small bowl, whisk the cornstarch with the beef broth until smooth. Pour the mixture into the skillet and stir to combine. Bring to a simmer and cook for 3-5 minutes, allowing the sauce to thicken.

Add the Cheese : Reduce the heat to low and sprinkle the chopped provolone cheese over the beef mixture. Stir gently until the cheese is melted and evenly incorporated.

Assemble the Sloppy Joes : Toast the brioche buns lightly if desired. Spoon the Philly cheesesteak mixture onto the bottom halves of the buns. Top with the remaining halves and serve hot.