Preheat oven or grill to 500°F.
In a large bowl, toss potatoes with 2 tablespoons of olive oil, salt, and pepper. Spread potatoes out in one layer on a parchment lined or non-stick baking sheet.
Roast or grill until browned on one side, about 9 minutes, then flip and roast on other side, until brown, about 5 minutes more. Remove from oven, and reserve 18 potato slices for garnish. Place the rest of the potatoes in a large bowl.
In a small bowl, whisk together the remaining 2 tablespoons olive oil, white wine vinegar, shallot, salt and pepper.
Top potatoes with arugula, and drizzle with vinaigrette. Toss well and serve, garnishing each plate with reserved potato slices and shaved Parmesan.