If your session has timed out and you click something in Toad that causes it to hang while waiting for the timeout to kick in, try this.
Click the X to close Toad
Then you should get this dialog, with the "try to restore" option.
Click "Try to restore the program". It actually works. Toad will realize that the connection is lost, it will come out of the "Not Responding" state, and then you can click Session -> Test/Reconnect to reconnect to the database.