Grease grill and heat over medium-high heat. Once hot, add patties and cover. Cook until brown and slightly charred on one side, about 4-6 minutes. Flip burgers (but do not press down), and glaze each burger with 1 tablespoon of the
gochujang. Cook an additional 4-6 minutes, topping each patty with cheese the last minute of cooking. Cover and cook an additional 1-2 minutes or to desired doneness. Remove to a plate and tent with foil and let rest 5 minutes.