Shakespeare and Broadway have had a relationship quite a bit more intimate than a handshake over the years, but the two have perhaps never met with quite the coziness that they do in “Broadway & the Bard,” an odd but enjoyable solo show, starring the stage veteran Len Cariou, that juxtaposes monologues and sonnets with songs from musicals. Longtime musical-theatergoers know Mr. Cariou best from his appearances in “Applause,” “A Little Night Music” and of course “Sweeney Todd,” in which he created the title role. But as he notes at the top of this 80-minute show at the Lion Theater, six months before appearing in “Applause,” he made his Broadway debut in 1969 in the title role of “Henry V.” His Shakespearean bona fides also include many seasons at the Guthrie Theater and at the Stratford Shakespeare Festival. (Mr. Cariou was born in Canada.) This bit of history explains one of the unlikelier transitions in the show, conceived by Mr. Cariou in collaboration with the director, Barry Kleinbort, and the music director, Mark Janas. Henry’s rousing speech at Harfleur (“Once more unto the breach”) is followed immediately by the title tune from “Applause.” There’s more sentiment than logic in this pairing, but most of the other segments are more cleanly aligned.