What is happening is that the Thunderbolt IP bridge only builds the link when the connection is first trained. When the other Mac reboots, its end of the cable drops, but the mini keeps the stale state and never retrains, so the service sits at Not connected until something forces the link to renegotiate. Unplugging the cable is the blunt way to force it. The software way to force the same renegotiation is to disable and re-enable the Thunderbolt Bridge network service, and you want to do it on the Mac that was restarted, since that is the side you can still reach:
In Terminal on the restarted Mac:
networksetup -setnetworkserviceenabled "Thunderbolt Bridge" off
wait a few seconds, then the same command with on.
Test it manually once after a restart. If it brings the connection back the way the cable pull does, you can automate it with a small LaunchDaemon on that Mac that runs those two commands at boot, and the connection will heal itself a few seconds after every restart.
Two smaller things worth checking while you are at it: use a short certified Thunderbolt cable (long passive cables are flaky about retraining), and try a different Thunderbolt port on the mini, the ports are not all wired through the same controller path. But the service toggle is the fix most people in your setup end up keeping, because the underlying failure to retrain on a peer reboot is Apple's to fix.
