Apparently, there's no way to use API Gateway with Cognito, it looks (and should be possible) there are so many lacks in the documentation... I'm following this forum thread [0] regarding Cognito integration in JS.
[0] https://forums.aws.amazon.com/message.jspa?messageID=645618#...
This is something I've also noticed. Even their official examples are really slow [0]
About auth: I've been researching API Gateway + Lambda and auth via Amazon Cognito. As of now the best example for auth I've found is LambdAuth [1]. I'm trying to add the API Gateway in the mix, it looks possible to integrate it.
I also agree that docs and more explicit examples are needed.
[0] https://aws.amazon.com/blogs/compute/the-squirrelbin-archite...
[1] https://github.com/danilop/LambdAuth