Gently clean the mushroom caps then rub each with a little Extra virgin olive oil and sprinkle both sides with salt and pepper. Grill over medium heat until just cooked through, a couple minutes a side. Remove from grill.
To serve, pour just enough dressing on the arugula to gently coat it. Pile the arugula on two plates. Slice the mushrooms and arrange over the arugula. Top each with a good amount of fresh shaved Parmigiano-Reggiano.