In reply to Tom Redmond: Some first routes could be:-
- Petite Aiguille Verte (NW ridge)
- Aig du Tour (take your pick!)
- Aig Argentiere (Mileu Glacier)
- Tour Ronde (East Ridge)
- Mt Blanc du Tacul (NW face)
- Cosmiques Arete on the Midi
I'm afraid I can't advise on current conditions, so some of the above might not be accessible or sensible.
1 axe should be fine.