Forget about FAA-JAA/EASA Conversions it's a dead end now, will cost you too much and will be a paperwork nightmare
That is not correct an I have an email form the head guru at the CAA to prove it.
Here is an option:
FAA training plus JAA/EASA ATPL exams in the US
Practical conversion in Europe.
PFT (pay for time) in either the US or Asia, that is one of teh reasons why you need to FAA CPL also.
So buy 300-500 hrs of turbo prop time where you fly as an FO, SIC/PIC or if you have the funds a typerating + 300/500 hrs of type, Airbus or 737.
12-14 months before right seat 737, 2 years to a paying job.
I personally know people that have done it that way and one of them is flying for Emirates as we speak.
Expect to spend around 100K Euro though, but it is t quick way and will save you a couple of years of "paying your dues" flying night freight and all sorts of other stuff at ungodly hours.
PM if you want more info.