Preheat the oven to 425 degrees F. Line a cookie sheet or large rimmed baking pan with parchment paper and lay the round of pie crust on top.
Toss the peaches and cherries with the cornstarch, vanilla, cinnamon and 1/3 cup sugar in a bowl until well combined. Mound the fruit mixture in the center of the dough round and spread slightly, leaving a 1 1/2-inch border. Fold the edges over the fruit, pleating as necessary. Whisk the egg and mix with a tablespoon of water. Using a pastry brush, brush the egg wash on the exposed crust. Sprinkle the remaining tablespoon sugar over the top of the fruit. Bake until the crust is golden brown all over and the fruit juices are thick and bubbling, 30 to 35 minutes. Cool slightly and serve warm or at room temperature.