Rolleys, there is no SIC type rating for a single pilot aeroplane under FAA. There is also no validation from the FAA on your EASA license. Basically to get any remuneration for operating a FAA registered aeroplane you need to have an FAA Comm. with the appropriate type rating. If you are not operating for remuneration, then it is private ops which then it is unlikely that you have an AOC which requires two pilots, and to operate it in JAR land as such would restrict you to Day VFR only, which means you would have to fly under the transition altitude all the time.
Basically the answer to your question is no, you cannot log the time at all, but you can ride along in the right seat for fun/ experience.