No, there are not any browsers that support this, but apparently it is not
very complicated to implement server-requested TLS upgrades in Firefox or
any another application using NSS:
The way I see it, Firefox would look for responses with HTTP 426 (Upgrade
Required) code and then the exchange specified in RFC2817 would take place.
I am not sure this will actually work :) One step at a time... :)
Apparently, it has been done before, but not documented.