If you're running Proxmox v6.x, then I assume that these are both v16.0 TurnKey releases? Also, I'm guessing that these are unprivileged containers? And you're setting a static IP for them from the Proxmox host when setting up the machines? Working on those assumptions:

We've had reports of Webmin being a bit unreliable and was hoping that the updated package(s) should resolve that. FWIW Webmin actually runs behind Stunnel (an SSL/TLS reverse proxy).

However, we're not had any reports of LigHTTPd (aka Lighty) having reliability issues. FYI Adminer in both the MySQL and PostgreSQL appliances is hosted by LigHTTPd.

I've not come across the 'PR_CONNECT_RESET_ERROR" error personally, but a quick online search suggests that it's usually caused by the server and the browser not being able to agree on a SSL/TLS certificate version/encryption algorithm.

In v16.x TurnKey servers (both Lighty and Stunnel) should default to TLSv1.3 but should also provide the ability to downgrade to TLSv1.2 (for some older browsers/OS). It's expected behaviour for that to cause issues with really old devices that use unsupported OS and/or browsers that rely on TLSv1.1 or earlier (e.g. IE on WinXP, early Android devices, etc). FWIW anything lower than TLSv1.2 is insecure. However, that should not be an issue with a modern browser on a supported OS. And it definitely should either "just work" - or not. It should never be intermittent!

So I have no idea how a TurnKey server could be doing this?! My suspicion is that either the error message is a misnomer (and something else is actually going on) or that the issue is actually being caused somewhere else?! Perhaps an IP clash (where you are connecting to a different machine)? Perhaps something weird going on with your Proxmox host? I'd also be inclined to suspect your guest browser and/or OS, but you note that you've tried multiple browsers and OS, so that seems unlikely...

I suggest that you try to recreate the issue and when it occurs, check the status of Lighty and/or Stunnel. To check the services, you can use 'systemctl', i.e.:

systemctl status lighttpd.service
systemctl status stunnel4@webmin.service

You can see the full history of these services since last reboot via the journal, i.e.:

journalctl -u lighttpd.service
journalctl -u stunnel4@webmin.service

The Stunnel process doesn't log to file by default, but Lighty should be logging to files in /var/log/lighttpd/. So you can also check them for errors. Please feel free to post back with any of these logs if you want a hand interpreting them.