Preheat oven to 375 degrees. Place puff pastry rounds in a single layer on a parchment-lined baking sheet and refrigerate while mushrooms cook. In a large skillet, heat oil over medium-high. Add shallot and cook, stirring, until soft, 3 minutes. Add mushrooms and cook, stirring, until soft and browned, 10 minutes. Season with salt and pepper. Stir in thyme and remove skillet from heat.
Lightly oil 6 jumbo muffin cups. Divide mushroom mixture and cheese among cups, then top each with a chilled pastry round. Bake until pastry is golden brown and puffed, about 25 minutes, rotating pan halfway through.
Run a small knife around cups to loosen tartlets. Place a rimmed baking sheet or large plate over pan and invert to release tartlets. Sprinkle with more thyme. Serve warm.