Helpfully, DNSMadeEasy provide us with an API along with documentation and a simple CLI wrapper for the API.
This project just runs that script inside a Docker container. This means you should be able to run something along the lines of the following in order to get a list of managed domains:
$ export DNSMADEEASY_API_KEY="key"
$ export DNSMADEEASY_API_SECRET_KEY="secret"
$ docker run --env DNSMADEEASY_API_KEY --env DNSMADEEASY_API_SECRET_KEY samcrang/dnsmeapi "http://api{.sandbox}.dnsmadeeasy.com/V2.0/dns/managed"