Mac at 100% CPU Usage when plugged into monitor
In the end, after I let Activity Monitor calculate things for a few minutes, it was Google Chrome Helpers causing the CPU load and not the connection to the monitor. I quit Google Chrome and killed its Helpers, then restarted Chrome about an hour later, and the problem fully subsided.

If the CPU load doesn't come from one of the apps, and really is only happening because of then monitor connection, see bmike's answer.