Heat oven to 375°F. Unroll dough on work surface (if using crescent rolls, pinch seams to seal); cut into 24 (2-inch) squares. Press 1 square in bottom and up side of each of 24 ungreased mini muffin cups.
In 10-inch skillet, heat oil over medium heat. Add onion and thyme; cook 8 to 10 minutes, stirring frequently. Reduce heat to medium-low; cook 6 to 9 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.
Bake 12 to 15 minutes or until golden brown. Remove tartlets from pan immediately. Serve warm.