Remove all outer leaves from the brussels sprouts (reserve cores for roasting or pickling).
Place the brussels sprouts in a large bowl with garlic, shallots, 2 tablespoons lemon juice, 1/4 cup Parmesan and season with salt and pepper. Massage leaves until they are all evenly coated. Let sit about 5 minutes. (This will help them soften.)
Add walnuts, apple and pomegranate seeds (if using) to brussels sprouts and toss to coat.
Add mint, parsley and olive oil and season with salt, pepper and more lemon juice, if desired.
Garnish with shavings of Parmesan.