Heat a large skillet over medium-high heat. Add the ground beef in an even layer and let sit undisturbed for 2-3 minutes to develop a brown crust. Break apart and continue cooking until browned throughout, about 5-7 minutes total. Transfer beef to a plate, leaving the fat in the skillet.

Add butter to the skillet. Once melted, add the diced onion, bell pepper, and minced mushrooms. Let cook without stirring for 1-2 minutes to caramelize, then stir and cook for another 2-3 minutes until vegetables are softened.

In a small bowl or measuring cup, whisk together the beef broth and cornstarch until completely smooth with no lumps.

Return the browned beef to the skillet with the vegetables. Pour in the beef broth mixture, ketchup, Worcestershire sauce, salt, and black pepper. Stir everything together thoroughly.

Let the mixture simmer over medium heat for 3-5 minutes, stirring frequently, until the sauce thickens and coats the beef and vegetables. The consistency should be saucy but not soupy.

Turn off the heat completely. Add the chopped Provolone cheese and stir continuously until it melts into the mixture, creating a creamy, gooey filling.

Toast the brioche buns until golden. For extra richness, brush the cut sides with butter before toasting. Pile the cheesy beef mixture generously onto the bottom buns, top with the other half, and serve immediately.