Preheat broiler with the rack 8-10 inches from the heat.
Core apples and cut into 1/2-inch slices. Whisk together olive oil, vinegar, mustard. Toss apples in this mixture and spread in an single layer on a large baking sheet. Broil until browned and soft or about 8 minutes.
Toss apples and all juices with mixed greens. Plate and serve immediately, topped with fresh parsley and toasted nuts.