Double-foil targets separated by a low density plasma and irradiated by a petawatt-class laser are shown to be a copious source of coherent broadband radiation. Simulations show that a dense sheet of relativistic electrons is formed during the interaction of the laser with the tenuous plasma between the two foils. The coherent motion of the electron sheet as it transits the second foil results in strong broadband emission in the extreme ultraviolet, consistent with our experimental observations.