I can strongly recommend the Stubai glacier in Austria, which is reasonably large. Very pretty area. In late May you should be able to ski down to about 2300m, which (from memory) is the mid station of the gondola, so you get quite good vertical.
Stubai is a short distance from Innsbruck, if you can get a flight there (it's a short distance from Innsbruck, even if you can't get a flight there).
My wife learned to ski on that glacier at May bank holiday, and I had plenty of fun too. A former Austrian racer, Heini Messner, used to run amateur race camps up there, which British tourists used to enjoy. Maybe he still does.
Stay in Neustift, a lovely local village in the valley, which has a postbus service up to the glacier (at least it used to).
Or there's a sport hotel at the base of the glacier which I stayed in once, and can recommend.
The Hintertux glacier is more extensive than Stubai, but would not be quite so easy to reach. I think you could drive there in around 2.5 hours from Munich airport.
I've also skied the two glaciers above Solden, which are not bad, and the Kitzsteinhorn above Kaprun (that's also a very pretty valley and resort).