I was able to solve this problem... The issue has something to do with the length of time the SMTP server is spending sending that email.
I'm not sure why, but it took more than a minute to send to that particular email address. During that time the connection to the database timed out.
The solution that worked was to close the connection to the database before sending the email and then re-establish a connection once the mail was sent. This appears to have solved the problem.