Heat a large skillet over medium-high heat. Add the lean ground beef and cook until browned, breaking it apart as it cooks. Remove the beef from the pan and set aside.
In the same skillet, melt the butter. Add the diced onion, green bell pepper, and minced mushrooms. Sauté until softened and nicely browned.
In a small bowl, whisk together the beef broth, cornstarch, ketchup, salt, and black pepper until smooth.
Pour the broth mixture into the skillet with the vegetables. Simmer for a few minutes, stirring often, until the sauce thickens.
Return the cooked beef to the skillet and stir well to coat everything evenly. This is where Gordon Ramsay Philly Cheesesteak Sloppy Joes get their rich, savory texture.
Turn off the heat and stir in the provolone cheese until melted and creamy.
Lightly butter the brioche buns if desired, then toast them in a separate pan until golden.
Spoon the cheesy beef mixture onto the toasted buns and serve warm.