Core and thinly slice the apple, unpeeled. Toss with lemon juice in a large bowl.
Remove outer layers of cabbages and use a large knife to thinly slice into ribbons. Add to apples.
Wash and grate (with the large holes) the raw beet. Add to apples and cabbage.
Whisk together the vinegars, mustard, and olive oil. Pour over the slaw and toss well. Season with salt and freshly ground pepper if needed. The flavors age nicely if you have the time to let this sit for an hour or two-- or more.