Trim the fennel bulb and reserve the fronds. Cut the bulb in half and remove the tough inner core. Using a sharp knife or a mandolin cut the fennel into very thin slices and put them into a large bowl. Cut the onion into very thin slices and add them to the bowl. Add the arugula to the bowl and give it a good toss. Pour in the olive oil and lemon juice, season it with salt and pepper, and toss it again to coat all the ingredients; put the salad onto a large platter. Using a vegetable peeler, shave thin pieces of the Parmigiano-Reggiano and scatter them over the salad. Garnish with the reserved fennel fronds.