A clogged converter will generally cause an engine to run rich. The converter is easily checked with a vacuum gauge but the problem there is that for some odd reason many mechanics do not use them or know how to interpret them.
Any chance this could be caused by a loss of residual fuel pressure? The system should hold pressure while the engine is at rest. If pressure is lost (usually a faulty check valve in the fuel pump) the reason for the engine running rough is because it’s taking time to burp the air out of the fuel lines. This is a common problem with many cars: it’s not Audi specific.
If the engine is up and running fine once warmed up you might shut it off and wait a few seconds before attempting a restart. If it restarts without stumbling then there could be a check valve problem in the pump.
This can also be verified by connecting a fuel pressure gauge and keeping an eye on it as the car sits for a while.
Hope that helps some.