If you want to focus on climbs in the Dolomites, consider staying in Cortina d'Ampezzo or in Corvara (the latter is right on the famous Sella Ronda loop). There are also famous Giro d'Italia and Tour de France climbs in Piedmont (NW Italy): try Bike Hotel Lo Scoiattolo in Pralormo, SE of Turin (Torino).
For the French Alps, apart from Annecy, St. Jean de Maurienne and Briançon also make good base locations.
I also do week-long bike tours in Piedmont, out of Chiusa di Pesio (near Cuneo, about equidistant from Torino, Genova and Nice); and in the Rhône-Alpes, out of Pont-en-Royans (not too far from Grenoble), as
Agile Compass Cycling Tours, but it's not only big climbs day in and day out.