Marek,
What training would you like to do?
In FAA land there are two types of flight training - Part 61 and Part 141. Part 141 is "an FAA approved school." But that doesn't mean Part 61 is bad. Part 61 is most flight training in the US. In most cases pilots who are not legal residents of the US must train at a Part 141 school that can process a visa application.
In the US, a Certificated Flight Instructor (CFI) is what EASA calls an FI(A). A CFI can have ratings for Airplane Single Engine Land, Airplane Multi-engine Land, Instrument, etc.
Simplistically, an "authorized instructor" is a CFI, with everything up-to-date, in an N-registered airplane. Foreign instruction "counts" for the most part.
A pilot with an EASA license can get a private pilot certificate issued under Part 61.75 that serves as the basis for further training. I believe that with your EASA ATPL you can get an FAA ATP rating. Do you have 1500 hours? You can do this in conjunction with a type rating. Others here will be much more knowledgeable about the process.
Hope this helps.
Terry