Step 1Place the juice from one lemon into a large bowl. While whisking, slowly add the olive oil to emulsify. Add sea salt and pepper to taste and set aside.
Step 2Place the remaining lemon juice in a medium bowl and fill with cold water. Peel the artichokes down to the tender core, and slice them lengthwise on a mandoline or as thinly as possible. Place the slices in the acidulated water to keep them from turning brown. Trim the fennel and also slice lengthwise, reserving in the acidulated water.
Step 3Shave the Parmigiano-Reggiano with a vegetable peeler (four pieces of shaved cheese per serving) and set aside.
Step 4Toss the arugula into the large bowl with the dressing. Drain the artichokes and fennel, pat dry and toss them into the bowl with the arugula. 5. Divide the salad among four salad plates and top with the shaved Parmigiano-Reggiano and parsley. Serve immediately.