Preheat oven to 350°.
Remove stems from mushrooms, and finely chop stems to measure 1/4 cup. Discard remaining stems. Combine 1/4 cup chopped stems, onion, and next 6 ingredients (through garlic).
Heat a large nonstick skillet over medium heat; coat pan with cooking spray. Add onion mixture to pan; cook 10 minutes or until vegetables are tender. Combine onion mixture and bread in a large bowl, tossing to combine. Slowly add broth to bread mixture, tossing to coat. Add feta; toss gently.
Remove brown gills from the undersides of mushroom caps using a spoon; discard gills. Place mushrooms, stem side up, on a baking sheet coated with cooking spray. Brush mushrooms evenly with 1 tablespoon vinaigrette. Sprinkle Parmesan and black pepper evenly over mushrooms; top each with 1/2 cup bread mixture. Bake at 350° for 25 minutes or until mushrooms are tender.
Combine remaining 2 tablespoons vinaigrette and greens, tossing gently. Place 1 cup greens on each of 4 plates; top each serving with 1 mushroom.