https://www.staubern.ch

This hut is located right between Hoher Kasten and Saxer Lücke. It's called Berggasthaus Stauber.

The location is perfect if you want to go see the Saxer Lücke before all the tourists come along for the day. Or even if you don't stay there for the night it's a nice spot to enjoy a "Nussgipfel", some kind of nut croissant, which tastes like heaven.