Okay a Peter Pan au no one asked for: 

Shinji’s an orphan; his mother died when he was very young and his father abandoned him a few years later. He gets adopted by a school teacher and his life is just kind of eh. 

But then Kaworu Nagisa (Peter Pan) loses his shadow and it finds refuge in Shinji’s room (we all know how this goes). Also, Kaworu’s lucky Shinji’s good with a thread and needle so he can attach the shadow back to Kaworu’s feet. 

Rei is Kaworu’s shy fairy that takes a while to warm up to anyone, but once she does she’s really caring. Instead of the Lost Boys, Kaworu’s gang would be this horrifying ensemble of all the Angels (Shinji finds them horrifying anyway), and Asuka would be Tiger Lily. 

Gendo’s the evil pirate villain, and the big reveal is that he abandoned Shinji to try and harness the magic of Never Land to bring Yui back.