Forum Moderators: not2easy
You'll need to set up this method in advance by saving a recovery token with your Facebook account. A recovery token is encrypted so Facebook can't read your personal information. If you ever need to recover your GitHub account, you can re-authenticate to Facebook and we will send the token back to GitHub with a time-stamped counter-signature. Facebook doesn't share your personal data with GitHub, either; they only need Facebook's assertion that the person recovering is the same who saved the token, which can be done without revealing who you are.
This can happen in just a few clicks in your browser, all over HTTPS. Facebook: Account Security Improvements With Delegated Recovery [facebook.com]