Arrange a rack in the middle of the oven and heat to 350°F. Spread the hazelnuts on a rimmed baking sheet. Toast until lightly golden-brown and the skins are peeling away, 10 to 15 minutes. Transfer the nuts to a clean dish towel, wrap up the nuts, and let steam for a minute. Rub the nuts with the towel to remove as many skins as possible. (Not all of the skins will be removed, which is fine.) Coarsely chop the nuts and set aside.

Shave the Brussels sprouts by gripping the stem end with your fingers and using a mandoline to slice as thinly as possible down to the stem, or until the sprout is too short to safely slice. (Alternatively, you can use a food processor to shave the sprouts. Trim and discard the stems and use the slicing blade of the food processor to finely shred the sprouts.) Place the shaved sprouts in a large bowl.

Halve and core the apple, then use the mandoline to cut into ¼-inch thick slices. Place the apple slices in a small bowl and toss with 2 teaspoons of the lemon juice to prevent browning; set aside.

Have a small Pyrex or other heatproof liquid measuring cup ready. Melt the butter in a small saucepan over medium heat. Cook, swirling occasionally, until butter is dark brown and smells nutty. Immediately pour into the measuring cup to stop the cooking. Let cool slightly.

Place the vinegar, remaining 2 teaspoons lemon juice, and salt in a small bowl and whisk to combine. While whisking constantly, slowly drizzle in the brown butter until the dressing is thick and emulsified. Whisk in the oil. Taste and season as needed.

Pour the dressing over the Brussels sprouts and mix in thoroughly with your hands. Add the apples and nuts and toss to combine.