hldj.org

Is there a way to point a shortcut to HLDJ and have it automatically execute a game?

No, but if you set "Auto-start default game" it will automatically execute the last game you started.
I suppose I can add this to the todo list, in the meantime if you're handy with scripting, you can make a script to update your settings.hldj to do what you want or edit the source: http://hldj.org/svn

This is now implemented and will be available in the next release.