Yes you can do all FAA training if you wish and then convert: (Is what I did)
To convert you need: (Pass ATPL EXAMS)
CPL (determined by your European Schoo, but in my case was 5 hours)
IFR/ME (and Class ME Rating) 15 hours (which 10 on sim and 5 real flight).
And checkrides for both and license emission fees.
So add 8000 euros aprox at todays date to convert everything from FAA to EASA.