Light, buttery puff pastry makes these turnovers a delectable treat. Pre-cooking the apples and spices caramelizes the filling, adding great flavor. This technique also works well for quinces or pears. Note that the apple filling can be made a day ahead and stored in the refrigerator until you assemble the turnovers.
Toss the apple slices with the lemon juice and place them in a large skillet. Add the sugar, spices, salt, and butter; cook over medium-high heat until the apples begin to soften but still hold their shape, about 5 minutes. Add the cream and currants and cook until the sauce thickens and the apples are soft and glazed with caramel, about 15 minutes. Remove the pan from the heat and cool completely.
Cut the puff pastry into six 5 x 5-inch squares. Divide the filling among the pastry squares, placing the apples on the bottom corner of each piece of dough. Leave a 1/2-inch border of pastry uncovered.
Brush the edge of the dough on all 4 sides of the pastry squares with some of the beaten egg. Fold the dough over the filling to make a triangle-shaped packet. Press the edges of the pastry together with the tines of a fork. Cut two small slits in the top of each turnover. Brush lightly with egg and sprinkle with sugar. Place the turnovers on a rimmed baking sheet lined with parchment or wax paper and transfer to the freezer for 15 minutes.
While the turnovers are chilling, position a rack in the middle of the oven and preheat to 400 degrees F.
Transfer the chilled turnovers to the hot oven. Bake until puffed and golden brown, 25 to 35 minutes.