Hard to believe you're climbing on scorching hot rock while looking up across the valley and seeing the snow capped peaks!
If the weather is bad at high altitude, or the snow level is down to 3000m then I would recommend the "Index" and the "Chapelle de La Gliere". Both can be climbed easily from the Index telesiege (which opens in the second week of June). They are excellent multipitch routes and the details can be found in either Gaston Rebuffat's "100 best climbs in the Mt Blanc Massive", or in The Alpine Club's guide book to the region.
Kinda cheesy signing a summit log for cragging the area but...its a great place to be. Where else can you climb a 4000m peak one day, crag in your shorts the next, and then do a 12 pitch outing on warm rock the next?