The EHA receives no signal from the computer when the engine is first started cold. Have a look at the lambda (or the voltage to the EHA). You will see it come to life after the O2 sensor is warm enough to provide some information for the computer to work with.

Or unplug the EHA. The car will start the same when cold, EHA unplugged or not.

The system is basically mechanical. The EHA only fine tunes the mixture. The engine will still run without the EHA's influence. It's a pressure fuel-injection system, just like on airplane engines - which have no electrical input at all, BTW.

__________________
2012 E350
2006 Callaway SC560