Well it doesn't sound like a serious fault although anything intermittent is a pain and when it causes the car to cut out can be dangerous. The way you describe it sounds just like a starter motor solenoid bad connection type issue......... except that wouldn't explain the cutting out.
It could also be an electronic issue with a sensor or the ECU but again that wouldn't explain the starter motor not turning over.. puzzling
Is it just the engine cutting out? or are all the electrics dieing?
The only thing I can think of at the moment is ignition switch contacts faulty / intermittent, this would be random though so if possible try to see if there is a definite pattern, for instance if it is always 15 mins before it will start again then that points more towards to something getting too hot, but working again when it has cooled down, or a possible fault with the immobiliser.
The ignition switch can crack / split inside but it usually just causes the key in the lock to be stiffer to turn and not to return to the "ignition on" position after stating rather than not actually starting, I've noticed an Audi sourced replacement ignition switch was different / improved internally to prevent this, so I think it's a known issue...... The part is cheep enough, (in the uk at least), but it's not a pleasant job to do due to lack of access, and the steering lock.
Need more detailed info really, next time it happens wiggle the key in the lock to see if that helps before trying to start it again.... also if you can locate where the click you hear is coming from is it inside the car, in the engine bay etc....