You can send the minidumps to vlk (at) avast (dot) com for analysis.
As a workaround, maybe it works to delay the update of avast into a permanent internet connection (DSL) you can change the value AlwaysConnectedWaitSeconds into avast4.ini file. Change (or add) AlwaysConnectedWaitSeconds value into the [InetWD] section of avast4.ini file. You could set the number of seconds to wait before attempting to update. The default is 30 but it may not be enough in some DSL systems. In this case, you can try increasing this e.g. to 60 (1 minute). That should be enough.
More details here: http://forum.avast.com/index.php?board=2;action=display;threadid=1647