This route isnt climbed so often. Not in the summer anyway. Its more of an escape route in bad weather which is a serious undertakeing in itself. Also if i wanted to choose a less frequent route, id choose something more interesting. Presuming you want to climb from France?
A brilliant route to the summit is to do the full traverse. Dome de miage - Bionnassay - Mont blanc.
It is usually done in three days. The route begins from Contamines-Montjoie, with the night spent in the Conscrits cabin. The following day, the Dômes de Miages is crossed and the night spent at the Durier cabin. The third day proceeds through l'Aiguille de Bionnassay and then the Dôme du Goûter.