In reply to EuanM:
There are dedicated, groomed ski mountaineering tracks in the MonteRosaSki area which sound as if they may be up your street:
https://www.visitmonterosa.com/en/experiences/slow/uphill-slopes-for-skimo/
the "skimo" tracks are marked in green on the piste map:
https://www.visitmonterosa.com/en/monterosa-ski/in-tempo-reale-winter/
If that gets you going then there are lots more "proper" freeride/ski touring routes in and around the Champoluc/Gressoney/Alagna valleys. I'm pretty certain that I dug up some useful links from when we went to Champoluc last year but I can't access them just now. I'm sure some judicious Googling would turn them up.
As it says on that web site, you can buy a "ski mountaineering ticket" for a day which gives you access to the uplifts you need for the full prepared route. However, if you have the area lift pass anyway you don't need the special ticket, and you can use any lifts in the area to get to the start of a "skimo" route if you're not based locally (eg from Champoluc you can use the lifts and the downhill pistes - including the "legendary" Sarezza 'goat run' - to access the start of the "skimo" track at the top of the Frachey-Alpe Ciarcerio funicular, which is more fun than taking the bus).
One small word of warning: some 'downhill only' skiers don't seem to understand the concept of a groomed slope reserved for people who like skiing uphill. I did spot one or two folks attempting to ski down the "skimo" tracks - I hope they found the flatter bits less than exhilarating and soon learned their lesson. (To be fair, encountering the occasional downhiller is probably no more risky for the ski mountaineer than skinning up the side of a proper piste.)
Geneva probably isn't the best place to start from to reach the MonteRosaSki area - Turin is better - but at least there's a tunnel under Mont Blanc to make access by road a bit easier!