So recently my galaxy s3's screen broke and I lost my Google authentactor app. I thought I had all the backup codes stored in my laptop but when I went to recover them I found out that one of my family members decided to go on a cleaning spree and deleted a bunch of files because they "didn't know what they for" So my question is there anyway to recover the codes off of my old phone?
You cannot recover codes from this app as it produces new ones every minute and doesn't provide Backup codes functionality. Perhaps, file recovery software might help you in recovering them from laptop but that's more like an opportunity lost here.
I always have my backup codes saved in a cloud in case any tragedy occurs.
you could flash a custom recovery to your phone such as Clockworkmod, reboot into recovery and from there you can use ADB to pull /data/data directory, in there would be the google authenticator app.
if you got another S3 you could flash a custom recovery and then push the /data/data directory(after installing the app again) and google authenticator would then work as it did on the old phone.
its possible to flash a custom recovery without a working screen as it uses the volume keys and the power button.