This riff on the class French Tarte Tatin is baked with oven-dried fruit.
Heat oven to 375 degrees. Line a baking sheet with parchment paper. Have ready six 4-inch fluted tartlet pans with nonremovable bottoms.
On a lightly floured surface, roll out the pie dough to slightly less than 1/8 inch thick. Cut out six 4 1/2-inch-round circles. Transfer the circles to prepared baking sheet. Cover with plastic; transfer to refrigerator to chill.
Place the sugar and 1/4 cup water in a small saucepan. Place pan over medium-high heat, and stir until the sugar has dissolved. Bring the mixture to a boil, and cook, uncovered, without stirring. As the water evaporates, the sugar will begin to caramelize. Once the sugar starts to brown, swirl the pan occasionally so the sugar caramelizes evenly. Reduce heat to medium. Cook until caramel is light brown, 3 to 4 minutes. Working quickly, divide caramel among tart pans, covering the bottoms evenly. Set the tart pans aside.
Arrange the fruit, cut sides up or down, on top of caramel in pans, covering bottoms.
In a small bowl, whisk together the egg and 1 teaspoon water. Remove the chilled dough from the refrigerator. Center the rounds over the fruit in the pans, laying the dough flat and not tucking it in. Using a knife, make a knife slit in each piece of dough. Brush rounds with egg wash. Transfer pans to refrigerator; chill 15 minutes.
Arrange the tart pans on baking sheet, leaving plenty of space in between. Bake until the crusts are golden brown, 20 to 30 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 tart onto a plate, tapping to loosen any fruit. Serve tarts warm or at room temperature.
Use any small dried fruit you like in these individual tarts; you'lll need enough so the fruit packs tightly.