Filling: Peel, core, and cut the apples into 1-inch pieces (about 4 cups). Put the apples in a medium saucepan, and add the water, sugar, and lemon juice. Bring to a boil, over medium-high heat. stirring occasionally, until the sugar dissolves. Cover, then reduce the heat to medium-low and simmer until the apples are very tender, stirring frequently, about 12 minutes. Remove from the heat. Gently mash the apples with fork or potato masher until the mixture is very soft, but still chunky. Cool completely. The filling can be made 2 days in advance. Cover and refrigerate.
Position 1 rack in the top third of the oven, and 1 rack in the bottom third, and preheat the oven to 400 degrees F. Line 2 baking sheets with parchment paper.
Pastry: If using a 14-ounce package (1 sheet), roll out the pastry on a lightly floured surface to 15-inch square. If using a 16-ounce package (2 sheets), stack the pastry sheets together, and then roll them out on a lightly floured surface to a 15-inch square. Cut the pastry into 9 (5-inch) squares. Put 1 generous tablespoon of filling in the center of each of 8 squares, reserve the remaining square for another use). Lightly brush the edges of each square with beaten egg. Fold half of the pastry square over the filling, forming triangles. Press and pinch the pastry edges with fingertips to seal tightly. Lightly brush the pastry triangles with beaten egg, and sprinkle lightly with superfine sugar, if desired. Using a thin, sharp knife, make 3 small slits on top of each triangle to allow the steam to escape. Put the triangles on the prepared baking sheets. Refrigerate until firm, about 15 minutes.
Bake the turnovers until they are beginning to color, about 15 minutes. Reverse the baking sheets from top to bottom. Reduce the oven temperature to 350 degrees F, and continue baking until the turnovers are firm and golden, 10 to 15 minutes longer. Cool at least 15 minutes before serving. Serve warm or at room temperature with and Caramel Sauce and vanilla ice cream.
Over low heat, mix the water, and sugar in a medium heavy-bottomed saucepan. Cook until the sugar dissolves, about 5 to 10 minutes. Do not stir. Increase the heat to medium and boil uncovered until the sugar turns a warm chestnut brown (about 350 degrees F on a candy thermometer), about 5 to 7 minutes, gently swirling the pan to stir the mixture. Be careful - the mixture is extremely hot!
Watch the mixture very carefully at the end, as it will go from caramel to burnt very quickly. Turn off the heat, and stand back to avoid splattering. Slowly add the cream. Don't panic - the cream will bubble violently, and the caramel will solidify.
Simmer over low heat, stirring constantly, until the caramel dissolves and the sauce is smooth, about 2 minutes. Allow the sauce to cool to room temperature, at least 4 hours. It will thicken as it sits.