HTTPS with ESP8266WebServer? I see many ESP8266HTTPClient examples, pasting in a fingerprint from an outside website, but I want to host an html page from the ESP8266 in AP mode using HTTPS.

Judging by the lack of examples, and the lack of a domain for a cert in AP mode, I'm thinking this is impossible, but would like confirmation before I give up.

Is it impossible to get the browser's green padlock on an AP-served web page?
The ESP8266 just doesn't have sufficient CPU power to do this reliably. 

It is possible, since the EPS8266 SDK docs have an "SSL User Manual" which describes this scenario.

In my experience, using the Arduino core for ESP8266, it doesn't work reliably.

Consider using an nginx reverse-proxy and having that proxy handle SSL negotiation. Understand that this is more for situations where the ESP8266 is local, in a network sense, to the nginx proxy, and you consider your WLAN secure enough.

On the plus side, you have room to grow in key sizes and/or complexity.