Heading to Europe this winter -- Any recommendations for the best ski area for advanced skiers in the Alps?

Oh you should definitely try Les Portes Du Soleil, it's across the Swiss/French border, and you've got no less than 6 ski resorts around, including the most famous Avoriaz.
The domain is just so huge!!!! And there are whatever slopes and jumps and forest paths you want! Brilliant - may I add I live in Switzerland and could be your guide :)

I really like Zuoz - it's a small village in Oberengardine, which is a valley relatively near to Austria.

It's a small area, with just three t-bars. But: because of the t-bars, not many people are there. So you don't have to line up and wait, or fight crowds, like you do in other areas. And the area is steep - which means it has long, challenging runs.

If Zuoz gets boring, there is also Lagalp and Diavolezza in the area, they are both also challenging, and have smaller crowds. They both have gondolas - which means if you are first out and fast, you get the run to yourself... or also if you wait for the others to go first!

You can see more about the village of Zuoz, and the area of Engadine, in the links below.

The very famous town of St. Moritz is also in Engadine - but don't be fooled. Their resorts, especially Corvatch and Corviglia, are for people who want to be seen skiing - and not for skiiers. You can hang & party in Mortitz in the afternoon / evening - but ski Zuoz, Lagalb or Diavolezza during the day!!

Have fun!