Use a mandoline or a sharp knife to slice the sprouts lengthwise as thinly as you can. Set them aside.
Peel and mince the shallot, and place in a large bowl. Stir in the oil, a tablespoon of lemon juice, zest and salt and pepper.
Add the shaved brussels sprouts and the dill, and toss to combine. Then add the Parmesan and taste for seasoning. You may need to add more lemon or salt to taste.