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Swiss side is nicer as you get the south facing views of the Alps. The French side is smaller roads (often with works on them) and you'll be in the shade.
If you'll be here after the 27th and want to ride Mauvoisin, give me a shout and you can drop your bags here (Chable) and I might ride it with you! I'd recommend taking the road on the north side of the Bagnes valley after Lourtier up to Mauvoisin, recently resurfaced and much better views.Have a look at the route the Tour went last year from Bern to Aigle, that was quite a nice route through the Bern countryside.
Excellent. Thanks all for the tips. I think I'm gonna head south out of Basel towards Thun, and then play it by ear from there. Either going down to the Rhone Valley (which involves getting a special train to transport me through a 15km tunnel!), or heading across towards Gstaad and Aigle. And given your recommendations @chiroshi, I'm guessing the Swiss side of Lac Leman is nicer than the French side?
I'm also tempted by taking a little detour up to the Mauvoisin Dam if my legs can carry me there. Maybe I'll try to dump my bags in Martigny.
Also the Swiss tourism website is so damn fancy! http://www.myswitzerland.com/en-dk/pages/listicle-collection/outdoor-listicles-collections/dead-ends-listicle.html
http://www.myswitzerland.com/en-dk/suggestions/road-cycling-in-switzerland/sanetsch-pass.html