The Picoweb do not support websocket. Even uasyncio on PicoWeb is old, it is not the official v3 uasyncio compatible, but it is good start point.
The
https://github.com/jczic/MicroWebSrv use WebSocket, but is not official
uwebsocket module - it wrote their own websocket module. And is better (if is possible) to use the official, for compatibility and so on.
This project
https://github.com/Carglglz/upydev/tree/master/upyutils has a WebREPL example working with SSL using the official uwebsocket module - that is great. This project was put as a PR (
https://github.com/micropython/micropython/pull/5611#) to support SSL in the official WebREPL, but I do not know why still not was merged, after two years.
I need just to know now how to add uasyncio on that WebREPL + SSL example.