It's not going to be easy, nor cheap and offers no respect of your experience, other than the fact that you have greater than the required 1500 hours.
Basically, it'll involve sitting 14 exams (and paying for the privilege of doing so), getting a class 1 medical from the country in which you're going to get the licence issued, doing any training considered necessary by the training organisation you choose to use for your type rating and getting a course completion certificate from them and then doing a licence skills test (LST) in order to get that type rating (from an EASA TRE). Finally, an English proficiency test.
So, in the UK, you would nip along to the CAA with:
14 exam passes (no Freedom Of Information, so you can't just study the answers)
UK class 1 medical
UK EASA TRE-signed LST
Course completion certificate
English certificate
A load of application forms for the licence and type rating to be put on that licence
Your credit card
You need to 'open' your licence with a type rating, hence the need to do one before the licence is issued.....they won't just issue you a blank ATPL with no type on it. Clearly, if your last type was a 744 then you'd probably choose to do a type rating on that but be very careful with which training organisation you use. Remember, you have to get a course completion certificate from them BEFORE you can sit the actual type rating test (LST). You can, therefore, imagine how some organisations take advantage and will require more hours in the simulator than others before they sign that certificate and, in the meantime, clear out your bank account paying for those hours......especially if you haven't got a sim partner.
I know a lot of US pilots complain about the FAA but be very prepared if you want to enter the bureaucratic world of EASA. It's all about forms, money and more forms and woe betide you if you haven't got one of those forms correctly signed!
My advice would be to forget the EASA licence idea all together. If you really want to fly (or need to fly) that badly in Europe then sign up with a bunch of agencies because sometimes the contracts will allow FAA licences; they could be flying 'N' registered aircraft, for instance. The other thing to consider is if you'd be able to work in Europe. You didn't mention it in your post but for most work in Europe you have to have the 'right' to do so i.e. do you have a EU passport (for most of EASA Land)?
NOW you can see how aviation-friendly the USA is
CAP 804 is the UK CAA reference for all things licensing.