In a large skillet, heat the olive oil over medium heat. Add the diced onion and green bell pepper, cooking until softened, about 5 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.
Add the ground beef to the skillet, breaking it apart with a spatula. Cook until browned and fully cooked, about 6-8 minutes. Drain excess fat if necessary.
Stir in the Worcestershire sauce, soy sauce, Dijon mustard, and beef broth. Season with salt and pepper to taste. Bring the mixture to a simmer and let it cook for about 10 minutes, allowing the flavors to meld and the sauce to thicken slightly.
Remove the skillet from heat and stir in the shredded provolone cheese until melted and well combined.
Serve the cheesy beef mixture on hamburger buns, and enjoy your delicious Philly Cheesesteak Sloppy Joes!