Brown ground beef in a large skillet over high heat. Drain fat. Stir in garlic, onion, and cabbage; cook until vegetables are wilted and soft. Stir in the beef broth, Worcestershire sauce, and mustard. Cook over medium heat, uncovered, until reduced to a sloppy joe consistency. Season with salt and pepper.
Place hamburger buns on top of meat mixture; cover skillet to steam, about 1 minute. To serve, fill steamed buns with beef and cabbage mixture.
