Yes you can.
That issue has been resolved by EASA, you need FAA and EASA licences/medical etc. if you want to operate outside France (just French issue EASA if only operated in France). FAA maintenance of not complex (not twin turboprop or jet or more than 5.7 tonnes - I think), FAA plus part M if complex.
All of the above applies to an EASA aircraft not operating commercially. Annex II aircraft are different.