Heat a large pan over medium-high heat and brown the ground beef until fully cooked. Remove and set aside.
In the same pan, melt butter and sauté the diced onion, bell pepper, and mushrooms until softened and lightly browned.
In a bowl, mix beef broth, cornstarch, ketchup, salt, and pepper.
Pour the mixture into the pan with the vegetables and simmer until thickened.
Return the beef to the pan and stir well to combine.
Turn off the heat and add provolone cheese, stirring until fully melted and creamy.
Butter and toast the brioche buns until golden.
Spoon the Philly Cheesesteak Sloppy Joes mixture into the buns and serve warm.