Put greens and chicken in a large serving bowl; set aside. Heat 1 tbsp. oil in a large frying pan over medium-high heat. Add mushrooms and garlic and cook until browned and fragrant, about 5 minutes (longer if mushrooms give off a lot of water).
Add remaining 2 tbsp. oil, the vinegar, and mustard, stirring to combine. Cook until mixture starts to boil. Pour over greens, tossing to coat. Top with onion, goat cheese, and pine nuts.