Heat a large skillet over medium-high heat until very hot. Add the lean ground beef directly to the pan without oil, searing it for about 1 minute to develop a deep crust. Once browned, use a spatula to break it up, cooking until fully browned. Remove the beef from the pan and set it aside, leaving the flavorful drippings in the skillet.

In the same pan, reduce the heat to medium and add a generous pat of butter. Once melted, toss in the diced yellow onion, green bell pepper, and sliced brown mushrooms. Sauté these vegetables for about 5-7 minutes, stirring occasionally, until they are tender and golden.

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

Pour the prepared sauce into the skillet with the sautéed vegetables, stirring to combine thoroughly. Let this mixture bubble for about 3-5 minutes.

Return the browned ground beef to the skillet, mixing it well with the vegetable and sauce concoction.

Off the heat, stir in the chopped provolone cheese, allowing it to melt into the beef and vegetable mixture.

While the cheese is melting, spread butter on the inside of each brioche hamburger bun. Toast them face down in a clean skillet over medium heat until golden brown and slightly crispy.

Load the toasted brioche buns generously with the cheesy beef filling and serve immediately.