It took me A WHILE to get into to Capaldi, but two moments from the back of season 9 convinced he was brilliant, those being his war speech at the end of The Zygon Inversion, and Clara’s death sequence from Face the Raven. They both just gave him so much emotion to work with and it keeps my eyes glued to the screen.

The Zygon Inversion speech hands down, but I would also say the "I see hatred" scene in Into the Dalek. That was my "yeah, this guy is the Doctor" moment for Capaldi.

How about

"The man that stops the Monsters!"

Not sure if a whole episode counts, but for me - beyond the speeches already mentioned - it would be the whole of Heaven Sent. That episode was art, and I struggle to see it working with anyone but Peter Capaldi in the role.

"You know what happens if I don't move on? More people die. There are kids living rough near here. They may well be next on the menu. Do you want to help me? Do you want to stand here stamping your foot? Because let me tell you something. I'm two thousand years old, and I have never had the time for the luxury of outrage."

I'd love to know whether that was in Sarah Dollard's original script or a Moffat intervention. It merges the Doctor's weariness and benevolence perfectly. I like The Zygon Inversion hugely, but I prefer series 10 Capaldi. The scene where he talks to the two Masters also has to be there. I only choose this one because it's more concise and focused.

Probably his speeches in the Zygon Invasion and The Doctor Falls

It was when he came back for Clara in Deep Breath for me.

I genuinely felt like this new Doctor, who was so harsh and angry, would leave her. But then he came back. Leaving her was just a ruse! He came back! He knew she'd do well, he had every faith in her! The Doctor through and through.

I'm not doing this because I want to beat someone, or because I hate someone, or because, because I want to blame someone. It's not because it's fun and God knows it's not because it's easy. It's not even because it works, because it hardly ever does. I do what I do, because it's right! Because it's decent! And above all, it's kind. It's just that. Just kind. If I run away today, good people will die. If I stand and fight, some of them might live. Maybe not many, maybe not for long. Hey, you know, maybe there's no point in any of this at all, but it's the best I can do, so I'm going to do it. And I will stand here doing it till it kills me. You're going to die too, some day. How will that be? Have you thought about it? What would you die for? Who I am is where I stand. Where I stand, is where I fall.