Shred Brussels sprouts into 1/16” shreds using a food processor or knife.
Mix shredded sprouts, pomegranate, apple, pecans, cranberries, and Parmesan in a bowl.
Whisk together vinaigrette ingredients, pour over the salad, and toss.
Let the salad sit for 30 minutes to soften the sprouts. Serve with extra Parmesan, pomegranate, and pecans.