Sending a testmail from a server with a configured exim is easy as pie.
From your usershell:
tell exim that we want to send a mail to an recipient and tell us about the details of the sending procedure
$ exim -v email@example.com
Tell exim about the sender, the subject and input some text.
From: firstname.lastname@example.org Subject: Foobar Text Text Text I like Pie!
You should see the details about the SMTP connection. You can return to your shell with