Shave the Parmigiano-Reggiano into paper-thin slices with a vegetable peeler and set aside.
Combine the fennel and mushrooms in a bowl. Whisk together the olive oil and lemon juice in a small bowl. Season with salt and pepper. Pour the oil mixture over the mushroom mixture and combine well. Mix in the arugula leaves.
Arrange the mushroom mixture on 4 plates and top with the shaved cheese. Drizzle with truffle oil if desired.