In a large skillet, cook the ground beef over medium heat until browned and cooked through, breaking it apart with a spoon as it cooks. Drain any excess fat.
Add the chopped onion, green bell pepper, red bell pepper, and garlic to the skillet. Sauté for 3-4 minutes, until the vegetables soften.
Stir in the beef broth, ketchup, Worcestershire sauce, mustard, salt, and pepper. Bring the mixture to a simmer and cook for 5-7 minutes, allowing the flavors to meld and the sauce to thicken slightly.
Once the mixture has thickened, stir in the shredded cheese and cook for another 2-3 minutes, until the cheese is melted and the beef mixture is creamy and well combined.
Toast the sandwich buns or hoagie rolls lightly in a separate skillet or under the broiler.
Spoon the Philly cheese steak mixture onto the toasted buns and serve hot.