I'd like to consider this Azure feature:
https://www.youtube.com/watch?v=HxXs9_ggBfg
But it won't be of much use to me if I'm unable to configure a custom domain for it. I'm having no problem with custom domains for standard App Service websites, but this is for containers.
I've tried searching for documentation, but I'm coming up empty.
--EDIT--
The technology I'm interested in is Web Apps for Containers; thanks to @sam-cogan for pointing this out to me.