ESNI is primarily a feature of the SSL implementation, not of the web server. As far as I'm aware, it's not in OpenSSL yet, but is likely to be added once TLS 1.3 has been finalized.
I have heard Googles boringssl already supports it, but I have found no actual evidence of that.
Cloudflare apparently does support it already for all their websites though. (or at least the ones that use cloudflare dns also)