Pass. side mirror heated up too much (almost too hot to touch!) and the driver's one heated briefly when I turned the ignition on, but then went cold and stayed cold.
Replaced both mirror glasses and the problem was solved. I suspect the little silver thermostats at the back of each mirror glass were broken.
The heat isn't constant in the mirrors - they cycle the heat. They warm up, hit their temperature limit, and the heat goes off again. When they've cooled down the heating comes on again. Each on/off cycle lasts about 10 minutes.
New glass is about £30 for driver's side / £25 for pass. side from an Audi main dealer, but I know they're cheaper from motor factors.