Preheat your oven to 400°F (200°C). Unroll the puff pastry on a floured surface and cut it into squares or circles, depending on your desired shape for the tarts.
In a skillet, heat the olive oil over medium heat. Add the chopped shallots and sauté for about 2 minutes until translucent. Then, add the sliced mushrooms and cook for an additional 5 minutes until they are tender. Season with salt, pepper, and thyme.
Place the pastry pieces on a baking sheet lined with parchment paper. Spread a layer of softened goat cheese on each piece, leaving a small border around the edges. Spoon the mushroom and shallot mixture on top of the cheese.
Brush the pastry edges with the beaten egg to create a golden, flaky crust. Bake in the preheated oven for about 25-30 minutes or until the pastry is puffed and golden brown.
Remove the tarts from the oven and let them cool for a minute. Serve warm and watch them disappear!
