is there no restrictions on owning a N-registered helicopter in Europe and flying it on a FAA PPL-H?
Actually, there's no restriction on flying an N reg on your local licence in the country which granted you the licence. Or gaining a piggyback Faa licence and flying it anywhere in Europe.
But no hire and reward (hence private owners) unless there's a really, really special reason (ie none available of that type in the country on that country's reg).
So I fly my N reg aircraft in the UK and Europe and I've never taken an FAA flight test or written exam. Simples.