In a large skillet over medium-high heat, add lean ground beef, allowing it to sear without stirring for about 2-3 minutes. Once browned, break it apart with a spatula and cook until fully browned and no longer pink, about 5 more minutes. Transfer the beef to a bowl and set it aside while preserving the flavorful drippings in the pan.

In the same skillet, melt 1 tablespoon of butter over medium heat. Add diced yellow onion, green bell pepper, and brown mushrooms, sautéing them for about 5-7 minutes until they are tender and golden.

Whisk together 1 cup of beef broth, 2 tablespoons of cornstarch, 1 cup of ketchup, and 1 tablespoon of Worcestershire sauce in a bowl until smooth. Season with kosher salt and ground black pepper.

Pour the prepared sauce over the sautéed vegetables in the skillet, stirring well to combine all ingredients. Allow to simmer on medium heat for 3-5 minutes until thickened.

Return the browned ground beef to the skillet, stirring it into the thickened sauce and vegetable mixture. Cook for 2-3 minutes.

Lower the heat and add sliced provolone cheese to the skillet. Gently stir until the cheese melts completely.

Butter the brioche hamburger buns on the inside and toast them in a separate skillet over medium heat for about 2-3 minutes until golden brown.

Load each toasted brioche bun with the cheesy beef and vegetable mixture and serve immediately.