I've heard of ecu's overheating problem on 05-06, bike will shut off because of bad ecu. But let the 06 owners shed some more light on this, bump for attention.
While this does sound like the dreaded ECU issue, I was under the impression that it only effected the 05 model and was rectified with the 06. If that is not the case, someone feel free to correct me.
Is the FI light illuminated? If so, please pull the codes and let us know what they are.