Preheat a large skillet over medium-high heat until hot. Add lean ground beef and sear for about 1 minute until a crust forms. Cook for 5-7 minutes until browned, then set aside.

Melt a tablespoon of butter in the same skillet over medium heat. Add diced yellow onion, green bell pepper, and minced brown mushrooms. Cook for 5-7 minutes until golden and fragrant.

In a bowl, whisk together beef broth, cornstarch, ketchup, Worcestershire sauce, kosher salt, and black pepper. Pour into the skillet with vegetables and stir to combine.

Let the sauce simmer for 3-5 minutes, stirring occasionally. The sauce will thicken and brown bits can be scraped from the skillet.

Return browned ground beef to the skillet and mix well. Off heat, fold in chopped provolone cheese until melted.

Butter the cut sides of brioche buns. Toast in a skillet over medium heat until golden brown for about 2-3 minutes.

Load the cheesy filling onto the toasted brioche buns and serve immediately.