Place Brussels sprouts flat side down and thinly slice. Alternatively, you could slice them on a mandoline. Toss in a salad bowl with 1/2 teaspoon of salt and set aside.
Combine the vinegar, honey, and Dijon together in a small bowl. Stir to combine, then slowly whisk in the olive oil. Taste and add salt as needed.
Toss the Brussels sprouts with about half of the dressing, then toss in the apples, nuts, and cheese. Add a few grinds of black pepper. Taste and adjust seasoning. Serve with remaining dressing on the side.