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 receipient@foo.bar
Tell exim about the sender, the subject and input some text.
From: user@your.domain.example
Subject: Foobar
Text Text Text
I like Pie!
Press Ctrl+d
You should see the details about the SMTP connection. You can return to your shell with Ctrl+c