Numerous ski resorts brand themselves as “Mont-Blanc.” Ironically, neither the Italian nor the French sides of Mont Blanc have any routes on the mountain’s real face.
On the French side, close to the town of Chamonix, there are four resorts: Chamonix, Argentiere, Les Houches, and Vallorcine. Courmayeur Mont-Blanc is a resort on the Italian side. Many locals believe that Courmayeur billed as “the sunny side of Mont Blanc,” offers superior skiing (including off-piste skiing). Purchase a Mont-Blanc Unlimited pass, which is suitable for the entirety of both the French and Italian sides, or purchase a pass for each resort separately (two-day pass minimum). Mont-Blanc Unlimited offers limitless access to Chamonix, Les Houches, Megève-Evasion Mont-Blanc, Courmayeur, and Verbier ski regions.
Sliding a groomed slope from Chamonix to Courmayeur is not possible at all!
Skiing between the French and Italian sides is impossible due to their lack of connectivity. Nevertheless, the “Mont Blanc” cable car from the Italian side will take you to the French side (over Mont-Blanc proper). It is only freeriding down, though, and you must drive to the cable car in the nearby village of La Palud because it is not at the Courmayeur resort.
However, it is not feasible to go in reverse (from France to Italy), as this requires passing via the Mont Blanc tunnel. For a car, the (quite astronomical) tunnel toll is 40 euros, or 50 euros round trip. However, the bus back is also included if you make a reservation with Guides Courmayeur. This is how you complete the renowned Vallee Blanche descent.
Skiing to the summit of Helbronner, presumably from the Aiguille du Midi accessible from Chamonix, is the only way to get to Courmayeur from Chamonix. Which through glaciated terrain is a challenging endeavor. The Aiguille du Midi to Helbronner lift is only available during the summer.
The Courmayeur and Dolonne cable cars transport you to the Courmayeur base station Plan Chécrouit inside the town of Courmayeur. On the trail map, this is obvious, but when you look at hotel sites on Google Maps, it is not at all apparent. Avoid skiing all the way to the town if you’re staying in Courmayeur proper since you’ll wind up in Dolonne and have to walk back. The Courmayeur cable car must be used to descend from Plan Chécrouit.
Driving is required to get to La Thuile, but once there, it may be enjoyable to ski Italy and France in one day. However, accomplishing that requires a few traverses, and having POMA lifts makes it less favorable to snowboarders (mainly if you take a bad turn). Be careful to do it on a clear day as well.