Heat 2 tablespoons oil in large skillet over medium-low heat. Add onion and vinegar. Cook 15 to 20 minutes or until tender and golden brown. Remove from skillet.
Heat remaining 2 tablespoons oil in same skillet over medium-high heat. Add green beans, mushrooms and thyme. Cook 4 to 5 minutes, stirring occasionally. Stir in onions and salt. Cook 1 to 2 minutes to until desired doneness. Top with cheese.