Heat a large skillet over medium-high heat until very hot. Add the lean ground beef without breaking it up for the first minute. Cook until browned all over, about 4-5 minutes, then break it apart. Remove from the pan and set aside.

In the same skillet, melt 2 tablespoons of butter over medium heat. Once melted, add the diced yellow onion, green bell pepper, and brown mushrooms. Sauté for about 5-7 minutes until golden brown and fragrant.

In a medium bowl, whisk together beef broth, cornstarch, ketchup, Worcestershire sauce, kosher salt, and ground black pepper until fully combined.

Pour the prepared sauce mixture into the skillet with the sautéed vegetables. Stir well and let it simmer for 3-5 minutes until the sauce thickens.

Return the browned ground beef to the pan, mixing it thoroughly with the vegetable and sauce mixture. Sprinkle in the provolone cheese and stir until melted.

Spread a little butter on the cut sides of the brioche buns and toast them in a skillet over medium heat for about 2-3 minutes until golden brown.

Spoon the cheesy beef mixture generously into the toasted buns and serve immediately while warm.