To fly and work in Europe you need an EASA licence. The FAA licence is pretty much useless here in Europe unless you find a job on N-registered plane based in Europe.
You could always get both by doing your course in the US and then convert your FAA CPL to EASA, provided you passed the 14 exams for the ATPL theory.
You could also get your FAA certificate after your EASA licence, but why woudl you?
If you have the right to live and work in the States (American citizenship or Green Card), then forget about Europe and move permanently to the US, get your licence and just fly.
Now, I got my PPL in the US, ATPL theory in Europe, time building in the US and then CPL-IR-ME in Europe.