1. Prepare the Oven and Baking Sheet:

• Preheat oven to 375°F (190°C).
• Line a baking sheet with parchment paper.

2. Cook the Mushroom Shallot Filling:

• Heat olive oil and butter in a skillet over medium-high heat.
• Sauté shallots until translucent, about 2-3 minutes.
• Add minced garlic and cook for 1 minute until fragrant.
• Stir in mushrooms and cook until they release their moisture and soften, 5-7 minutes.
• Season with salt and pepper.
• Remove from heat and let the mixture cool slightly.

3. Prepare the Puff Pastry:

• Roll out the thawed puff pastry on a lightly floured surface to thin it slightly.
• Cut into equal-sized squares or rectangles.
• Arrange pastry pieces on the prepared baking sheet.

4. Assemble the Tarts:

• Spoon the mushroom shallot filling onto the center of each puff pastry piece.
• Crumble herbed goat cheese evenly over the filling.

5. Fold and Brush:

• Fold the edges of the puff pastry over the filling to create a rustic tart shape.
• Brush the exposed pastry with the beaten egg for a golden-brown finish.

6. Bake:

• Bake in the preheated oven for 20-25 minutes, until the tarts are golden brown and puffed.

7. Garnish and Serve:

• Garnish the tarts with fresh thyme leaves.
• Let cool for a few minutes before serving.