Own domain to AWS services

I wonder if is possible, and how to do it most simply, using my domain to request AWS API.

At this moment request to, e.g., Cognito
looks like: https://cognito-idp.eu-west-1.amazonaws.com/
I wonder to change it to sth like: https://cognito.mydomain.com/

Sth like a proxy for those services.

How to do in the best way?

Hmm for Cognito User Pool there might be a way to do this for certain API calls. But AFAIK it is only for the case where Cognito hosts your login page for you - https://docs.aws.amazon.com/cognito/latest/developerguide/cognito-user-pools-assign-domain-prefix.html