Phone [Q] Rescue S3 with broken display and digitizer and without USB debugging

Hi guys!
I have a Galaxy S3 with broken display and (MHL tested) broken digitizer.
Device works, I get home screen but can't unlock.

I would rescue the smartphone and I think that the way is using ADB (with droid@screen, screencast or similar).
I don't have alternatives right?
The problem is that I can't use ADB cause actually it does not have USB debugging enabled.
Just a couple of questions:
Can I use ADB in recovery mode (without USB debugging)?
With MHL I get broadcast in home screen, not before, is that normal? How about MHL in recovery mode?

I can only imagine a solution: Go in recovery mode and execute a "no-look" change of ROM installing something with USB debugging option enabled by default.
Do you have other solutions?
Who can help me doing this? Does exist a "how-to" somewhere? Which ROM?