S4 intermittent starting problems.

[Duckman]

My 2007 S4 (V8 manual transmission) has intermittent starting problems.  There seems to be no rhyme or reason when or why, but the car has plenty of electrical power, and the starter works perfectly when it works.  When this happens, it is after I've been driving the car, turn it off for a few minutes, and then try to restart it.  The key turns, all the electrical power activates, including lights, radio, power top, etc.  It is not that the motor turns over but won't catch.  Quite simply the starter does not engage.  

I thought it might have been the clutch starter switch, but I replaced it and it didn't fix the problem.  Repeated attempts to isolate the problem to fix it have failed.  There are no fault codes when it occurs, and there are no historical fault codes.  Nada.

The last thing I want is to pay a mechanic, especially not at the Audi dealer shop rate, to search and eliminate possibilities.  Has anybody experienced anything like this?  Does anybody have any great ideas of where the problem may lie?