Hi Avgeekstan. I was or are in the same position as you. I did the FAA way and got all the way to CFI with COM ME/SE IR and so on. It gave me a lot of flight time and an awesome experience but as you. I dont have greencard so Europe is the way. I should be done with the conversion process to EASA by October t his year. As rudestuff said it is possible but it is a complicated and long way, that requires a lot of research and is not for the faint of heard. If you want to know about my experience you are more than welcome to send me private message and we can connect.
Cheers!