Is this possible via the API? Is there already a website that does this?
Add "filter:verified" to the search query should work.
From: http://socialmeditation.wordpress.com/2013/04/30/filter-twitter-searches-by-verified-users/
If you are interested in using DataSift for this, here is an example live stream of Tweets from all verified Twitter users - http://datasift.com/stream/22288/tweets-from-verified-users
Just a thought...
I'd imagine that you can make a call to any user account and have it return all its followers, right? If this is true, then all you need to do is check against Twitter's Verified account found here: https://twitter.com/verified.
DataSift allows you to do this. It's Twitter's official partner for advanced searches.