What is the best way to get a count of how many times users have used the Facebook Connect login button (this thing) to login to a site?
I have read through https://developers.facebook.com/docs/reference/api/ to no avail. I don't have access to the Facebook API set up yet, but perhaps that provides a count of logins?
If you have your og tags set up properly:
fb:app_id - A Facebook Platform application ID that administers this page.
More info on that here: https://developers.facebook.com/docs/opengraphprotocol/
The Insights tab on your facebook app will give you the information you seek. Keep in mind though, that the data may be 48 hours behind.