During my two weeks in Chamonix we did the Cosmiques arête twice. It is a much nicer and more interesting way of getting back up to midi from col du midi than just walking up the ridge on the east side of midi. And, you get your two minutes of fame each time you reach the snow ridge just below the viewing platform on Aiguille du midi.
This route is a mixed climb graded AD- and probably takes two to three hours from the start below the Cosmiques hut depending on your speed.
Start up the ridge, mostly keeping a little bit to the right hand side of the ridge. Since this is a very popular route there will probably be quite well tread track to follow, at least on the snow covered parts.
Most of the route as seen from pointe lachenal.
The first abseil anchor is at the top of the first snowy pinnacle seen in the pictures. The first abseil is easily down-climbable which brings you to the second abseil. This one is about 15 m. From the rocks at the bottom continue horizontally to the east and then climb some near vertical rock with holes drilled for crampons and continue more or less horizontally.
One of us down-climbing the first abseil.
From here there are two options. We did both during our stay in Chamonix. If the snow is solid, it is easier to continue more or less horizontally to go around the next rock tower on the south side. If the snow is soft and/or unstable, instead countinue up to the ridge on mixed ground and go between two big rock towers to reach the snowy platform just below the crux of the route.
The platform before the rock. The lower section of the rock is the crux, graded 4a.
From the small platform at the top of the crux section continue to the right up a huge crack. From the next platform traverse to the north side of the ridge following a diagonal rock gully to reach the snow arête at the top of the big rock tower. From there, the viewing platform at Aiguille du midi is easily reached by a ladder.
Bring some medium sized friends and nuts to place some protection along the more exposed parts if you feel you need it. The whole ridge is simul-climbed (or short-roped), even the crux. A lot of people even solo it.
At least a 30 meter rope is necessary for the abseils.