In a large bowl combine half of the fresh or dried oregano, the Worcestershire sauce, salt, and black pepper. Add ground beef; mix well. Shape meat mixture into four 1/2-inch-thick patties; set aside.
In a large cast-iron or other heavy skillet heat 1 tablespoon of the oil over medium-high heat. Add mushrooms, sweet pepper, and onions; cook for 4 to 5 minutes or until crisp-tender, stirring occasionally. Remove from heat. Stir in the remaining fresh or dried oregano. Remove vegetables from skillet; cover and keep warm.
Add the remaining 1 tablespoon oil to skillet; heat over medium-high heat. Add patties. Cook for 12 to 15 minutes or until an instant-read thermometer inserted in the centers registers 160 degrees F, turning once. (Reduce heat if patties brown too quickly.)
To serve, fill toasted rolls with burgers, vegetables, and cheese.