Place the ground beef in a skillet and set the heat to medium-high. Continuously stir the beef until it is thoroughly browned. Upon completion, drain any excess grease and return the beef to the skillet.
Introduce the Worcestershire sauce and garlic powder to the skillet. Season with salt and pepper according to preference. Incorporate the diced onion and bell pepper to the mixture and add butter. Maintain the mixture on medium heat, allowing it to cook for an additional 5 minutes.
Proceed by adding the beef broth to the skillet. Increase the heat to bring the mixture to a boil. Once achieved, reduce the heat to a simmering point. Allow the mixture to simmer for approximately 6-8 minutes or until the liquid volume is halved.
Remove the skillet from the heat source. Evenly distribute slices of provolone cheese atop the beef mixture. Cover the skillet and allow it to stand undisturbed for 2-3 minutes, ensuring the cheese is completely melted. Following this, gently integrate the melted cheese with the beef mixture.
Serve the Sloppy Joe Philly Cheesesteak mixture in appropriate bread rolls or buns.