I assume you did not see a "check engine" light or record a code. One possibility is the mass air flow sensor, although this would usually trigger a code. You could disconnect the MAF connector and see if the car will idle. If MAF sensor is bad (or has a loose connection, etc) the car will idle with it disconnected but will probably still produce very little power.

If you disconnect the MAF sensor and your diagnostic system is working, the light will come on for that reason and you will need to reset it afterward. If the MAF is not connected, as well as for up to a couple of minutes at startup, the engine would normally run "open loop" - that might explain why you could restart after cooling down.

Did you check for a vacuum leak, such as a cracked or loose air intake hose (from the air filter)? This might or might not set a code.

Another possible issue is low fuel pressure/bad or intermittent fuel pump. I don't think this would set any code.

Many turbo owners have posted regarding performance issues that are specific to problems in the air and/or oil plumbing on the turbo.