In reply to rriding:
There's the refuge de la Pierre à Bérard which you can walk to from near Vallorcine. The approach is a pleasant, not too steep but fairly long walk. No snow or glacier but from the hut itself you can't see the Mont Blanc range.
The Albert Premier has a safe ascent in summer. You can take chairlift up behind Le Tour and the path contours across then up to the hut, with a steep climb just before the hut - splendid views of Mont Blanc and numerous snowy mountains.
The Torino hut involves a very scenic, and expensive, trip by telepherique across the Vallée Blanche, or a telepherique ride up from the Italian side... That could be good pretext for taking the tunnel and visiting Val Veni... There is a hut called the Elisabetta up there, with an easy approach, no glacier, but I haven't been to it myself - a wild area, quite different to the Chamonix valley with views of the Freney Pillars and the Aiguille Noire de Peutery - very impressive.