These tartlets are rich, sweet, and savory.
Heat the oven to 400 degrees. Line a baking sheet with parchment. Have ready six four-inch fluted tartlet pans with nonremovable bottoms. Using the large holes of a box grater, grate the cheese into a large bowl. Set aside.
Place 1 sheet of puff pastry on a lightly floured work surface. Without rolling out the dough, cut out 4 circles. Transfer the circles to the prepared baking sheet. From the second sheet of puff pastry, cut out 2 more circles; transfer these to the baking sheet. Cover the dough with plastic, and transfer the baking sheet to the refrigerator to chill.
Arrange the tartlet pans on a clean work surface. Drizzle 1 teaspoon olive oil into the bottom of each pan. Sprinkle half of the chopped herbs in an even layer over the oil. Arrange the dried tomatoes on top of the herbs, evenly covering the bottoms. Sprinkle with the reserved grated cheese and the remaining herbs. Season with salt and pepper to taste.
In a small bowl, whisk together the egg and 1 teaspoon water. Remove the chilled dough from the refrigerator. Center the rounds over the tomato filling in pans. Brush rounds with egg wash. Transfer tartlet pans to the freezer, and chill 15 minutes.
Remove the tartlet pans from the freezer, and transfer to the baking sheet. Bake until pastry is golden brown, about 20 minutes. Transfer baking sheet to a wire rack until pans are cool enough to handle.
Using your fingers, gently unstick crusts from rims as needed. Turn out each tartlet onto a plate, tapping to loosen any tomatoes. Serve tartlets warm or at room temperature.
You can also use Flaky Pie Dough for these tartlets instead of the puff pastry.