It must be a local process sending or receiving email, really. Let's see which one it is, maybe it will be an interesting discovery.
First, turn on verbose logging for the Mail Scanner. To do this, add the following line to the [MailScanner] section of <avast>\data\avast4.ini file:
Log=20
Then restart the Internet Mail provider.
Next, wait for the icon to reappear (i.e. wait for the connection to be made). Finally, go to <avast>\data\log, open the file aswMaiSv.log, and find a line with something like
--POP command REDIRECT
followed by some numbers. The last number on the line is the PID (Process ID) of the process that's making the connection. Use the Processes tab of the Task Manager to find out which process corresponds to this PID (enable the PID column first by using the View -> Select Columns command).
Hope this helps,
Vlk