Preparation

• Heat a large skillet over medium-high heat. Add the lean ground beef and break it up. Cook until browned and no longer pink. Season lightly with salt and pepper. Remove the beef to a bowl and set aside.
• In the same pan, reduce heat to medium and add the butter. When melted, add the diced onion, green bell pepper, and minced mushrooms. Sauté until everything is browned and any liquid from the mushrooms has evaporated — about 6–8 minutes.
• In a small bowl, whisk the beef broth with cornstarch and ketchup until smooth. Season with a pinch of salt and pepper. Pour this mixture into the pan with the vegetables. Bring to a simmer and cook until the sauce thickens and becomes glossy, about 2–3 minutes.
• Return the browned ground beef to the pan. Stir to combine and let the flavors come together for a minute on low heat.
• Turn off the heat and immediately stir in the chopped provolone until it melts into the meat and sauce. The residual heat melts the cheese without making it oily. Taste and adjust seasoning.
• Separately, spread a thin layer of butter on the brioche buns and toast them in a clean pan or under the broiler until golden.
• Spoon the cheesy beef mixture onto the toasted buns, close, and serve warm.