Thaw puff pastry as directed on package. Meanwhile, in a 12-inch nonstick skillet, heat 2 tablespoons of olive oil over medium-low heat. Cook and stir the onion for 15 to 20 minutes or until brown and caramelized. Stir in the brown sugar and vinegar.
Cook and stir for 2 minutes longer. Remove the onion mixture to a medium bowl and stir in ¾ cup sour cream, cheese, thyme, salt and pepper. Set aside.
Heat the oven to 400 degrees. Spray 24 medium muffin cups with nonstick spray. Wipe out the skillet with a paper towel. In the same skillet, heat the remaining 1 tablespoon olive oil over medium heat.
Cook and stir mushrooms in oil 5 to 8 minutes or until all of the liquid from mushrooms evaporates. Stir the mushrooms into the sour cream mixture.
On a lightly floured surface, roll each thawed puff pastry sheet to a 12-inch x 9-inch rectangle. Cut into 4 rows by 3 rows. Gently press the squares into the muffin cups.
Divide the mushroom mixture among the muffin cups. Bake for 12 to 15 minutes or until golden brown. Let stand for 1 minute before removing from the pans. Garnish each tart with a dollop of the remaining sour cream and serve immediately.