In reply to Trossachs:
I've posed this question here before a long while ago, I think the general thinking was along the lines of that there might be lots of snow still and lots of the huts could still be closed.
You could try Slovenia, I think when we went it was around that time and the huts were open and snow wasn't a problem. Does it have to be the Alps? Or is just European good enough?
As always, I would highly recommend Northumberland.... yes, it might be cold and raining, yes it might be boggy, yes you might need to run bits of the same, but I can guarantee solitude and silence! :-P