Turns out we had a very similar issue as described here:
Our external legacy application that connects to our Discourse API is running on an old Windows 2008 R2 server.
For whatever reason, the Windows server and the Discourse server were unable to agree on a cipher suite after the recent Discourse updates were installed earlier this week. Whether some ciphers were altered during the update, or if this issue coincided with a LetsEncrypt cert renewal at the same time, I don’t know 
Anyway, rather than edit our Discourse, I was able to add a couple of cipher suites to the Windows server that they both agreed on, again with the help of the SSL Labs link that @Falco shared above 