Set the "from" header in the email to be in the format "Full Name"
You can use hook_mail_alter() to change the "from" email address of any email sent out from your site. The email address used needs to conform to the RFC standard, which it does is your question's title. But user names can be tricky and might contain illegal characters. There's a comment on the D6 docs that provides a code sample for properly formatting the email address. Your code would look something like...

function yourmodule_mail_alter(&amp;$message) {
  $message['from'] = $message['headers']['From'] = '"Full Name" &lt;email@address.com&gt;';
}


Also, keep in mind that just setting the "from" address may not have the desired results in all email clients (see the discussion on the Drupal issue queue). You might also need to set "sender", "errors-to" and "reply-to" based on your requirements. Those values are in the "header" of the $message variable from the code sample above.