Migration Assistant (going from 2011 MBP running Mavericks to 2013 rMBP running Mavericks) broke Apple Mail
Turns out I'm not the only person this has happened to.  From this Apple Support Community Post, if you quit Mail, uncheck the checkboxes from the Internet Accounts prefpanes for anything Mail related, and then delete the files in the Container folder as mentioned in the posting, everything seems to work.