Chef Server (and Chef Automate) don't support running behind reverse proxy using URL routing

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(Note that I looked; https://discourse.chef.io/t/changing-the-chef-server-name-and-or-url/5180 and https://discourse.chef.io/t/changing-chef-api-url/6813 were asked by the same person 3.5 and 5 years ago and no one responded)

Chef Server Version

[What version of Chef Server? Any plugins? Are you using Chef Backend? What versions?]
Chef Server, Chef Automate (latest versions as of Feb 2019)

Platform Details

[Operating system distribution and release version. Cloud provider and instance type if running in the cloud. Total/Free RAM and disk space. Is this running in a container?]
Running on RHEL / CentOS on EC2 instances on AWS

Configuration

[Standalone, Tiered, HA, Chef Backend? New install or upgrade? Ideally include your /etc/opscode/chef-server.rb or otherwise provide details of changes from the default]
Chef and Chef Automate are both running as non-HA

Scenario:

[What you are trying to achieve and you can't?]
We have a reverse proxy that only supports URL based proxying (i.e. we'd like https://myserver/chef to proxy to https://chef.internal and https://myserver/automate to proxy to https://automate.internal). There does not appear to be any documentation on how to do this; unlike gitlab or other tools where you can either specifcy an external URL or otherwise specify a subdirectory for the application to listen for, chef and chefautomate appear to only support a hostname (FQDN).

Steps to Reproduce:

[If you are filing an issue what are the things we need to do in order to repro your problem? How are you using this cookbook or any resources it includes?]
Using stock configuration

Expected Result:

[What are you expecting to happen as the consequence of above reproduction steps?]
https://myexternalproxy/chef brings up the chef login page and all URL's are of the form /chef/ instead of just /

Actual Result:

[What actually happens after the reproduction steps? Include the error output or a link to a gist if possible.]
Fails miserably; redirects to /

Start by reproducing the stock Chef Server configuration behind a URL-based reverse proxy, using the /chef path and the redirect behavior described here. Trace the server's routing and URL generation, then verify that the login page, redirects, and generated links retain the /chef prefix rather than redirecting to /.