The FAA do not grant an approval to a non FAA pilot to fly an 'N' reg aeroplane in a single country (state in formal parlance) only.
What I
think you are saying here is not true. The FAA permits a US registered aircraft to be flown by the holder of a non-US licence only in the state that issued that licence. Since the EU is not a state, a French issued EASA licence entitles the holder to fly a US-registered aircraft only in French airspace. If I have misunderstood your statement, I apologise.