Get CloudFront custom domain in the headers of a request

Question

I have a CloudFront distribution abcd1234.cloudfront.net and I've added a custom domain of mysite.com to the distribution.

• The CloudFront distribution's origin: aaabbbccc.execute-api.us-east-1.amazonaws.com
• When I load the page mysite.com/hello/world, CloudFront is then consuming API Gateway aaabbbccc.execute-api.us-east-1.amazonaws.com/prod/{proxy+}
• the API Gateway path endpoint is invoking a Lambda Function that calls a function like getPageContent(customDomainName, pagePath) which should be mysite.com and /hello/world respectively.
• However, inside that function, the Host header that eventually makes it into the function's event.headers.Host value is never the custom domain. Instead, the Host header is always aaabbbccc.execute-api.us-east-1.amazonaws.com.

I want headers.Host to equal mysite.com (or another header to show that the request comes from mysite.com, but no matter what I do, the Host value is always just the origin url.

Edit: I tried whitelisting Host and it caused the site to break completely, with the error about not being able to reach the CloudFront distribution.