Nop,
We know for sure that if you want to fly a N registered aircraft/rotorcraft, also if it is based in Europe, you need a FAA License.
To do that a conversion is possible with a consolidated exam, a bunch of hours and a flight check, maximum one week and you have done.
But now, EASA (European Aviation SCREWING Agency) have generated a new rule, stating that every pilot non-EASA, could not fly inside the EASA environment with their license. And, you know well, the FAA/CASA/TC to EASA/JAA conversion, isn't as simple as the JAA to EASA/TC conversion.
Cheers,
PE