In a large bowl combine beef, Worcestershire sauce, half of the oregano, salt and pepper (try not to overmix). Shape into four 1/2-inch thick patties. Set aside.Advertisement
In a large cast-iron skillet heat 1 tablespoon of the oil over medium-high heat. Add mushrooms, onions and sweet peppers. Cook and stir for 4 to 5 minutes or until crisp-tender. Remove skillet from heat. Stir in remaining oregano. Transfer vegetable mixture to a medium bowl; cover to keep warm.
Return skillet to stovetop; add remaining 1 tablespoon oil. Heat over medium-high heat until oil is very hot. Add beef patties. Cook patties for 12 to 15 minutes or until an instant-read thermometer inserted in centers registers 160°F, turning once. (Reduce heat if burgers brown too quickly.)
Serve burgers topped with vegetables and feta on toasted buns.