How do I change the nameserver in a Microsoft Azure DNS Zone to be something else? I'd like to use Cloudflare for my nameservers instead of the default Azure ones.
As far as I know, you can not change Microsoft azure DNS name server. Azure DNS is a hosting service for DNS domains that provides name resolution by using Microsoft Azure infrastructure. But Azure DNS supports co-hosting domains with other DNS services.
To set up co-hosting, modify the NS records for the domain to point to the name servers of both providers. Read here.
For example, you can add name servers from Cloudflare in the additional name servers
in the NS record on the Azure portal.