Sure, both the ClientHello and ServerHello (which are the first two messages during setup) can contain arbitrary extension fields; these are already used for things like SNI, ALPN, "early data" (optimistic 0-RTT data), and so on. You could define an extension in the private-use range for your esoteric use.

I don't think there's any way to see this from Javscript, though. At least, not web-browser Javascript.