Brown the Beef:

1. In a large skillet over medium-high heat, cook the ground beef until browned, breaking it up with a spatula as it cooks. Season with a pinch of salt and pepper. Drain excess fat if necessary.

Sauté Vegetables:

1. Add the thinly sliced onion, green (and red, if using) bell peppers, and minced garlic to the skillet. Sauté for about 5 minutes until the vegetables soften and become slightly caramelized.

Create the Sauce:

1. Stir in the ketchup, Worcestershire sauce, Dijon mustard, and beef broth (if using). Allow the mixture to simmer for 5 minutes to meld the flavors.
2. Reduce the heat to low and sprinkle the shredded provolone cheese evenly over the mixture. Stir until the cheese melts into a creamy sauce. If a creamier sauce is desired, stir in the heavy cream and heat through.

Assemble the Sloppy Joes:

1. Toast the hamburger buns in a toaster oven or under the broiler until lightly crisp.
2. Spoon a generous portion of the cheesy beef mixture onto the bottom half of each bun. Add any optional toppings (like sautéed mushrooms or an extra slice of cheese) and top with the other half of the bun.
3. Serve immediately for a bold, flavorful dinner with a creative twist on the classic Philly cheesesteak.