Thaw the pastry and on a floured surface, roll it out to ½-inch thickness. Cut out 4 rounds, each 5 inches in diameter. Transfer the rounds to a baking sheet lined with parchment or waxed paper and refrigerate for at least 25 minutes.
Peel, core, and finely dice 1 apple (or pear). Place it in a small saucepan with ⅓ cup sugar and half the butter. Cook over low heat until the apple is very soft and the sugar is dissolved to make a spreadable compote. Set the pan aside to cool.
Preheat the oven to 350°. Cut four 6-inch squares of parchment or waxed paper. Arrange the squares on a baking sheet and sprinkle 2 tablespoons of sugar evenly over the sheets. Place the pastry rounds on top of the sugar and prick the entire surface of the dough with a fork.
Spread a tablespoon of cooled apple compote over each pastry round. Sprinkle the rosemary evenly over the compote. Peel, core and thinly slice the remaining apples (or pears). Fan out the slices in a circular pattern on top of the tarts.
In a small saucepan, melt the remaining butter. With a pastry brush, carefully coat the tops of the tarts. Sprinkle the remaining 2 tablespoons of sugar over the tarts.
Bake for 18 minutes, or until the apples begin to caramelize, turning light golden brown. Serve hot on dessert plates, dusted with powdered sugar and garnished with a mint sprig.
Makes 4 individual tarts