Using a mandoline with a hand guard, thinly shave the Brussels sprouts, discarding the tough ends. Thinly slice the endives and combine with the Brussels sprouts in a large bowl.
In a small bowl, whisk together the lemon juice and the oil. Season with salt and pepper, to taste. Drizzle the vinaigrette over the salad and toss to combine. Add the Pecorino and toss. Taste, and season with additional salt, pepper, or lemon as needed.