ubuntu, dns, vpn
So I discovered this today, and figured I'd pass it along in case anyone else is running into the same problem.
Before fixing it this morning, I'd connect to my (work) VPN and would be unable to resolve internal hostnames on the remote network. Turns out there was a regression in network-manager, and an older version was re-pinned to bionic-updates. If you've having this problem, "update" network-manager to version 1.10.6-2ubuntu1.1 (apt install --reinstall network-manager=###)
ubuntu, dns, vpn
It'll technically be a downgrade (it'll say as much in the console) but when it's done, and a reboot has occurred, DNS will work as expected on your VPN connection.
Full commands because of character limits:
apt install --reinstall network-manager=1.10.6-2ubuntu1.1
apt install --reinstall network-manager-config-connectivity-ubuntu=1.10.6-2ubuntu1.1
ubuntu, dns, vpn
There are a few other packages that also appear to be related (and which I "updated" the same way):
gir1.2-nm-1.0=1.10.6-2ubuntu1.1
libnm-glib-vpn1=1.10.6-2ubuntu1.1
libnm-glib4=1.10.6-2ubuntu1.1
libnm-util2=1.10.6-2ubuntu1.1
libnm0=1.10.6-2ubuntu1.1
Although I must admit that everything started working for me after just downgrading network-manager. The other packages I only downgraded "just in case".