Heat outdoor grill or cast iron grill pan. Gently brush off any dirt from caps. In a small bowl whisk together the oil and vinegar and season with salt & pepper. Brush caps with vinegar & oil mixture and place on grill. Cook for 2 minutes per side or until mushroom is tender. Brush occasionally with vinegar and oil. Drizzle remaining vinegar and oil over arugula. Shave parmesan into greens and toss. Slice grilled portobellos and arrange on top of arugula