Right now I believe the only option to fly for a European airline on an FAA certificate is with Norwegian. The only bases available for that are Stateside, FLL for the 787 or PVD/SWF for the 737.

If you wish to fly in Europe, you will need an "unlimited right to live and work in Europe", which means citizenship* in an EU/EEA country or Switzerland, plus an EASA license. On top of that, most European carriers expect you to have a type rating in an aircraft which they operate, and they do not provide that type rating without some sort of training bond.

*- I say citizenship because for all intents and purposes, that is the only way to have unlimited right to live and work all throughout EU/EEA/CH. A temporary or permanent residence permit only allows for the right to live and work in one specific country, not all of Europe.