Shinji actually coming out on top during his conversation with "himself" in the 10th Angel, at times easily picking apart its attempts to break him by talking, and at times turning the tables on it.
Leliel!Shinji: So why risk making people hate you? Shinji: You mean, with the banter? Well... the Doctor does that all of the time. And people never seem to care. They just think he's funny. Leliel!Shinji: Maybe. But we're not the Doctor, are we? Later in the conversation Leliel!Shinji: So we managed to inspire pity in a man who wants to help literally everyone. Not much of an accomplishment. Shinji: Yeah, maybe. Or... maybe... he saw something in us that even we can't see. Something worthwhile. Leliel!Shinji: Like what? Shinji: I don't know. It's like you said. We're not the Doctor.
Realising that Leliel!Shinji is actually afraid:
Shinji: Oh. I see. Leliel!Shinji: Somehow, I doubt that. Shinji:You're afraid, too, aren't you? You're afraid of making mistakes. Leliel!Shinji:(Silence. Shows seen of a future Shinji being considered insane)
He also figures out that Leliel!Shinji isn't really him, pointing out that, despite Leliel!Shinji's claims that he was told but repressed the information, he didn't know about Asuka's mother or about Rose, and he wouldn't just repress that, nor could he just pick up on it subconsciously. And that it keeps going from telling him he sucks to telling him he should be God(I am not making that last one up)
How does the Doctor deal with the 11th Angel? He gets in the way of its beam and shows it his memories, so it knows what he has done and can do. The last we see of it, it's running away from Earth (or more accurately, running away from The Doctor) at 20% of light speed. And it's accelerating