Heat a large skillet over medium-high heat for about 2 minutes. Add the ground beef, breaking it up with a spatula as it cooks. Stir and let it brown evenly for about 5-7 minutes.

In the same skillet, reduce the heat to medium and add 2 tablespoons of butter. Once melted, stir in half diced and half minced onion along with diced green bell peppers. Sauté for about 5 minutes until tender.

Return the cooked ground beef to the skillet. Mix together the beef broth and cornstarch to form a slurry, then pour this mixture into the skillet. Cook for another 2-3 minutes until thickened.

Add in ¼ cup of ketchup, 2 tablespoons of Worcestershire sauce, kosher salt, and black pepper to taste. Allow to simmer for 3-5 minutes to meld the flavors.

Remove the skillet from heat and stir in the shredded cheese until fully melted.

Heat another pan over medium-low heat, add a tablespoon of butter, and toast each brioche bun cut-side down for about 2-3 minutes until golden brown.

Spoon the cheesy mixture onto the toasted buns. Serve hot, garnished with desired toppings.