Sauté the Veggies
Heat olive oil in a large skillet over medium heat. Add the diced onion and green pepper. Cook for 3–4 minutes, stirring occasionally, until soft.
Brown the Beef
Add the ground beef to the skillet with the vegetables. Break it apart with a spatula and cook until fully browned, about 5–7 minutes. Drain any excess fat if needed.
Add Seasonings and Sauce
Stir in ketchup, Worcestershire sauce, garlic powder, salt, and pepper. Cook for 1 minute to combine the flavors.
Thicken with Flour and Broth
Sprinkle flour over the beef mixture and stir to combine. Pour in beef broth and stir well. Let the mixture simmer for 2–3 minutes until it thickens.
Melt the Cheese
Lower the heat and lay provolone slices over the beef. Cover the skillet with a lid for a minute to melt the cheese, then stir until the cheese is fully incorporated and creamy.
Assemble the Sloppy Joes
Spoon the cheesy beef mixture onto toasted hamburger buns. Serve hot.