Combine apples, cherries, and currants in a medium heatproof bowl. In a small saucepan, heat apple juice until simmering and pour over dried fruit. Stir to combine, and let stand until juice is absorbed, about 5 minutes. Stir in applesauce; set aside.
Place puff pastry sheet on a lightly floured work surface. Trim edges, and cut out twelve 3-by-3-inch squares, discarding excess dough. Place a tablespoon of apple filling in center of each square. In a small bowl, stir together egg and 1 tablespoon water. Using a pastry brush, lightly brush the edges of each pastry square with egg wash. Fold pastry over filling to enclose, forming a triangle. Lightly press edges to seal. Using a floured fork, crimp edges decoratively with tines, if desired. Transfer turnovers to a parchment-lined baking sheet. Repeat with remaining pastry and filling. Chill for 30 minutes.
Preheat oven to 375 degrees, with rack in center. Brush turnovers lightly with egg wash, and sprinkle with sugar. Bake until deep golden brown and puffed, about 25 minutes. Let cool 10 minutes. Serve warm or at room temperature.