I don't see how you get the password hash in the first place
generally they send you a nonce you generate a nonce use them as salt and then send hashed pw + your nonce to facebook login server they take stored pw and perform the same operation and check if the results match then let you logon. at no point do they send you the password, a hash therof or any info about the password as for pw recovery, well generally they just e-mail you a new one there's no pw hashes exchanged for that.
how the hell would this work?