Preheat the oven to 400°F. On a piece of parchment paper, roll out one of the pastry sheets until it is about 14 x 9 inches. Transfer to a baking sheet. From the second sheet of pastry, cut enough 1½-inch-wide strips to form a border around the rolled-out pastry sheet. Brush the rim of the large rectangle lightly with water. Arrange the strips around the rim of the pastry sheet to form a border. Cover and refrigerate.

Using a sharp knife, make a series of slits in the apple quarters along the peeled sides, taking care not to slice all the way through. Place the apples in a microwave-safe bowl, add 2 tablespoons water, and cover with plastic wrap. Microwave for 3 minutes; drain. Combine the sugars and the cinnamon in a small bowl. Sprinkle half of the mixture over the center of the pastry shell. Place the apples over the sugar, slit sides up. Sprinkle the remaining sugar over apples and scatter the butter pieces all over. Brush the rim of the tart with the beaten egg.

Bake the tart until nicely browned, about 25 minutes. If the rim browns before the apples are cooked through, cover it with foil. Brush the jam over the apples and rim and serve warm.