In Episode 8, the Doctor discovers the existence of Lilith, and is furious. Later on, he ends up having a tense confrontation with Gendo Ikari. The Doctor wants answers about what he's discovered, and Gendo calmly offers an explanation for each of the former's arguments. The Doctor begrudgingly concedes, even though he doesn't approve of NERV's methods.
Later, the Doctor admits that Gendo was one step ahead of him the entire time. Gendo may not get the Father of the Year award in any way except ironically, but there's a reason he's got a reputation as a manipulative genius.
At the end of Episode 9, Gendo reveals that he plans on pitting the Doctor against the Seele Council.
While facing Israfel, Shinji has no idea what to do, so he asks himself what the Doctor would do. This results in Shinji running away... only to come running back at over Mach 1.2, tackling one half of Israfel through a stadium. Then a skyscraper.Then hitting it with his own Eva's detached arm so hard it bounces off the street and into a mall!
Israfel was defeated conventionally. No dance competition, no synchronicity exercises, no Berserker State, just Shinji letting himself cut loose in the Eva and Asuka doing the same.
It's a little more complicated than that. Throughout the story, the Doctor's been encouraging the pilots not only to think for themselves, but to think strategically rather than just rushing into things. Shinji's victory comes about as a result of him planning ahead and breaking a water line to flood a street before the battle even starts. He later lures his half of Israfel into the resulting mess and electrocutes it as a distraction at a key moment. Shinji doesn't just win by cutting loose; he wins with science.
Rei gets one by weaponizing the Reiquarium against a Weeping Angel. Recall that part of what makes it creepy is the unblinking stares of the Rei Clones? What's the one way to freeze an Angel in place? And she figured this plan out in seconds.
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.
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.
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.
