Caddy (currently in beta) will issue and renew SSL certificates automatically with no downtime (on Linux; Windows has very brief downtime during restarts).
I can confirm that Caddy + Let's Encrypt is the most seamless and awesome way to run TLS. I did this last night for one of my LE beta whitelisted domains and it took MAYBE 4 minutes. Caddy did all the work. Kudos to the Caddy team for such a great admin experience.