I found one other Broadway credit for him, from 1999, something called Not About Nightingales. It had a short run.
The New York Public Library site has a lot of pictures from the 1964 production of Hamlet, obviously because of Richard Burton's involvement. There may be pictures of Giannini among them but he is not identified by name and I couldn't tell if any of the men depicted were him or not. It was interesting that All My Children veteran actress Eileen Herlie played Gertrude.
There is a DVD available of Hamlet but IMDb does not list him as a cast member.