So sorry to see that. Basi is sensitive to cold. You did say the snow was unexpected, but it's best not to put basil outside until all danger of frost is over and overnight low temperatures have settled to mid-50's or above.
That said, if the plant was vigorous before, and any portion of the basal stem is undamaged, Insuppose it's not entirely impossible for it to re-grow new side shoots. I would remove the damaged leaves but not the stems unless it is mushy (cut back to solid stem), and don't uproot it just yet.