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Discard the outer leaves of the cabbage, then shave it as finely as you can with a sharp knife or mandoline, discarding any thick ribs as you go.
Place in a large bowl, drizzle over 1/4 cup oil and toss. If the salad seems a little dry, toss through some more oil. Allow to stand for at least 15 minutes, or up to a couple of hours.
Toss through the cheese and season with salt and pepper.
Arrange on a serving platter or in a salad bowl and drizzle over a sparing amount of balsamic.