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.