Broadway is getting a tryout on HBO and its sister digital network, HBO Signature. Both will telecast “Direct From Broadway,” a series of live-on-tape Broadway shows, including “Jekyll & Hyde: The Musical” starring David Hasselhoff, “Putting it Together: A Musical Revue” starring Carol Burnett, and “Smokey Joe’s Cafe: The Songs of Leiber and Stoller.” The productions will be telecast in December and January, premiering on HBO Signature and replaying in high definition on HBO.