Remove fronds, stalks, and tough outer layer from fennel. Halve bulb lengthwise and cut into paper-thin slices with a mandoline or other manual slicer. Whisk together broth, lemon juice, oil, sugar, and salt and pepper, to taste, in a large bowl. Add fennel, arugula, and onion, then toss together. Thinly shave enough Parmesan with a vegetable peeler to measure 1/4 cup.
Serve salad sprinkled with Parmesan.