In large skillet, heat oil over medium heat. Add onion and thyme; cook 8 to 10 minutes, stirring frequently. Reduce heat to medium-low; cook 10 to 15 minutes, stirring constantly, until onions are caramelized. Stir in brown sugar and wine; cook 2 to 3 minutes, stirring constantly, until liquid is gone. Stir in mushrooms. Spoon about 1 tablespoon onion-mushroom mixture into each cup. Top with walnuts and cheese.