Architect(arc.codes) and Begin.com(a service built on top of Architect) are awesome. They take much of the pain out of deploying web apps to AWS. But neither currently support multiple domain on a single app. Setting this up manually is painful. There is a (somewhat outdated) checklist on the Architect site that I used as a starting point. Here are the steps that worked for me.
Create a new Origin certificate for your domain on Cloudflare.
Add that Cloudflare Origin certificate to the AWS certificate manager.
Create a custom domain in the AWS API gateway with that certificate.
Map that AWS custom domain to your specific AWS API gateway endpoint.
Point your Cloudflare DNS to the API Gateway domain for the custom domain you created.
That is it. It may take a few minutes for the DNS to resolve.
Subdomains to different endpoints
Multiple Domains on the same arc endpoint
Upgrade Requests to HTTPS