I would wait for things to thaw a bit more before I tackled the valve.
I wonder if the water is passing through the valve body. I think that may be an option on some two handle tub/shower faucets. I've never plumbed on that way, but it seems some have that option.
Can you post a picture of the valve, 800 pixels or less?
Where pipes run by air ducts that allow cold air movement, I've seen pipes freeze before.