To prepare: Preheat the oven to 375 F. Place the onions in a shallow roasting pan and drizzle over the oil. Season generously and drizzle over half the balsamic vinegar. Roast for 25-30 minutes, stirring halfway through, until the onions are softened and browned.
Drizzle over the remaining balsamic vinegar and allow the onions to cool to room temperature. Using a vegetable peeler, shave the Parmesan into wafter-thin slices.
Arrange the arugula, roasted onions, and cheese on serving plates; drizzle with the pan juices and serve.