In a saucepan, cook the cherries over low heat, stirring, until they soften and most of their liquid has been exuded, about 10 minutes. Drain the cherries in a fine sieve, pressing lightly to extract as much juice as possible; reserve the cherries separately. There should be about 2 cups of juice; if necessary, add 1/4 cup of hot water. Return the juice to the pan, add the granulated sugar and lemon juice and bring to a boil.
Meanwhile, in a small bowl, stir the cornstarch into the water until dissolved. Add the cornstarch mixture to the cherry juice, stirring constantly. Add the cooked cherries, bring to a boil and cook until very thick, 2 to 3 minutes. Stir in the vanilla, transfer the cherry mixture to a bowl and refrigerate until chilled.
On a lightly floured work surface, roll out the pastry 1/8 inch thick. Using a 7-inch saucer or bowl, cut out 10 rounds; if necessary, stack the scraps, reroll them and cut out more rounds. Refrigerate the pastry rounds until firm.
Preheat the oven to 375°. Line a large baking sheet with parchment paper or foil. Working with 1 pastry round at a time, brush the edge with the egg. Spoon a scant 1/2 cup of the filling in the center of the round and fold the dough over to make a half moon. Press the edges to seal and crimp the seam with a fork. Transfer the turnover to the baking sheet. Continue to brush, fill and shape the remaining turnovers. Refrigerate until firm, about 10 minutes.
Cut three 1-inch-long slashes on each turnover. Brush the tops with the egg and sprinkle with the turbinado sugar. Bake the turnovers for about 35 minutes, or until the pastry is golden and the filling is bubbling. Transfer to a wire rack to cool before serving.
The baked turnovers can stand overnight at room temperature. Recrisp in a 325° oven.