With a very sharp knife, cut the Brussels sprouts, kale, and radicchio across in thin shreds and put them all in a large bowl.
In a small bowl or measuring cup, whisk together the lemon juice, olive oil, salt, and pepper. Pour enough dressing on the slaw to moisten it well. Add the cranberries.
Shave the Parmesan in big shards with a vegetable peeler, add it to the salad and toss well. Check for seasonings and serve cold or at room temperature.
The slaw can be dressed a few hours before serving but to prepare a day or two ahead , prep the salad and the salad and the dressing separately and combine few hours before serving.
Recipe courtesy of Ina Garten, All Rights Reserved, 2013