The present disclosure is directed to a nozzle cooling system for a gas turbine engine. An impingement plate is positioned radially inwardly from a radially inner surface of an inner side wall of a nozzle. The impingement plate and the inner side wall collectively define an inner chamber. The impingement plate includes a first portion defining one or more impingement apertures and a second portion defining one or more post-impingement apertures. A duct plate encloses the first portion of the impingement plate. The duct plate, the first portion of the impingement plate, and inner side wall collectively define an outer chamber in fluid communication with the inner chamber through the one or more impingement apertures. Compressed air from the outer chamber flows through the one or more impingement apertures into the inner chamber and exits the inner chamber through the one or more post-impingement apertures.