Tank you for the comment! I´ve added some information on how to get there (also see the two links I suggest at the bottom of the mounatin page). Unfortunately, I couldn´t find out about the hut, not even on the Mittenwald homepage, which lists several huts in the area. Sorry!
"So I was sitting in my cubicle today, and I realized, ever since I started working, every single day of my life has been worse than the day before it. So that means that every single day that you see me, that's on the worst day of my life."
--Peter Gibbons (Office Space)