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 the seasoning and serve cold or at room temperature.