EASA will be an ICAO recognized entity when it has a membership - which it doesn't. The Chicago convention which forms the basis of ICAO treaty recognizes sovereign states independently. What they otherwise do between themselves is their own business.
On that basis, as an ICAO member, FAA rules allow you to pilot an N-registered aircraft in a foreign nation using a license from that nation, but not the license of an otherwise politically associated neighboring nation.
The Postal History of ICAO