In a large skillet over medium heat, drizzle oil and sauté the onion and green bell pepper until soft (about 5 minutes).

Add the ground beef to the skillet and brown it (6 to 8 minutes), breaking it apart as it cooks; drain excess fat if needed.

Stir in garlic, Worcestershire sauce, ketchup, salt, and pepper. Simmer on low heat for an additional 5 minutes.

Turn off the heat and sprinkle cheese over the mixture; cover with a lid for about 2 minutes to melt.

Toast hamburger buns lightly in another skillet or under the broiler.

Spoon generous portions of the cheesy beef mixture onto each toasted bun half and serve hot.