How I fixed VPN DNS failures after Windows laptops woke from sleep

I dealt with this recently in an internal help desk queue where remote staff used Windows laptops, a corporate VPN, and split DNS. The visible problem was users could reconnect to VPN after sleep, but internal hostnames stopped resolving until they rebooted. The easy explanation was a bad VPN password or flaky Wi-Fi, but it appeared when the laptop slept while docked and then moved to home…