Preheat the oven to 400 degrees F.
Coat the bottom of a 9×9-baking dish with the olive oil, and then evenly distribute the grated Parmesan over the olive oil. Sprinkle with the salt and pepper.
Place the potato halves, cut side down, on top of the cheese mixture and bake for 30-35 minutes or until the potatoes are very tender when pierced with the tip of a knife. Precise baking time will depend on the thickness of your potatoes.
Cool for five minutes, and then remove to a plate with a metal spatula. You may have to gently cut between the potatoes with a knife as the cheese becomes crusty and may connect the potatoes to one another.
Place the arugula in the baking dish and toss with the olive oil and Parmesan that remains in the pan. Transfer the arugula to a serving platter or individual plates, and top with the roasted potatoes. Top with additional Parmesan and snipped chives, if desired.