In a server's "Manage SSL Certificate" settings (under Virtualmin's "Server Configuration" menu), I clicked the "Copy to Webmin" button and Webmin promptly stopped working over https (e.g. https://example.com:10000). This appears to be an unfortunate bug.
When hitting the usual URL for Webmin (https://example.com:10000), the miniserv error log says:
[28/May/2009:10:47:54 +0000] [xx.xx.xxx.xx] Bad Request
The /etc/webmin/miniserv conf file pointed to the virtual server's SSL key and certificate files in /etc/webmin (and they were indeed copied to the /etc/webmin directory). I edited the miniserv.conf file to point to the original SSL key and certificates and restarted Webmin and this fixed everything.