I am a little confused, and this was all several years ago so may well have changed...
When I converted an ICAO licence to FAA I had to sit the FAA theory, obtain the relevant medical, do 3 hours with an FAA CFI and then take a skills test.
When I converted an ICAO licence to JAA I had to sit the JAA theory, obtain the relevant medical, do a few hours training "at discretion of the head of training" and then take a skills test.
Seems they both have fundamentally the same requirements
In fact, the FAA had another step which involved getting a "restricted PPL" beforehand.
Unless I was fleeced and there was a different way to do the FAA conversion
EASA is not looking promising IMHO