Right away I figured I've got malware on the machine. I did the following:
Everything came up clean -- no detected problems.
Since the dns burst happened every day, I was trying to figure out how to determine what process on a Windows 7 machine issued a given dns query. And in the middle of working on that, some google searches pointed out that Avast itself is the culprit. Arghhhh.
Apparently Avast performs a dns lookup on the top 1000 sites to spot dns hijacking. Of couse, in doing so, Avast creates a highly suspicious traffic signature. Sounds like many people have wasted hours trying to hunt this down, only to find that Avast is the root cause. :(