Prepare the Filling

1. Heat the Pan: In a large skillet, heat olive oil and butter over medium heat.
2. Caramelize the Onions: Add the sliced onions and cook, stirring occasionally, for 15-20 minutes until golden and soft. Sprinkle with sugar and continue cooking for 5 minutes. Add balsamic vinegar, stir well, and remove from heat.
3. Cook the Mushrooms: In the same pan, sauté the mushrooms until they release their moisture and turn golden brown, about 8-10 minutes. Season with thyme (if using), salt, and pepper. Combine the mushrooms with the caramelized onions.

Prepare the Tartlet Shells

1. Preheat the Oven: Preheat your oven to 375°F (190°C).
2. Roll the Puff Pastry: On a lightly floured surface, roll out the puff pastry to about 1/8-inch thickness.
3. Cut the Dough: Use a round cookie cutter or a glass to cut out circles slightly larger than your tartlet molds.
4. Assemble the Tartlets: Press the pastry circles into the tartlet molds or muffin tins. Spoon the onion and mushroom mixture evenly into each shell.
5. Apply Egg Wash: Brush the edges of the pastry with beaten egg to give them a golden, glossy finish.

Bake the Tartlets

1. Bake: Place the tartlets in the preheated oven and bake for 20-25 minutes, or until the pastry is puffed and golden brown.
2. Cool Slightly: Allow the tartlets to cool for a few minutes before removing them from the molds. Serve warm or at room temperature.