Make the pie crusts, omitting the step to divide the dough into two discs. Wrap the dough with plastic wrap and refrigerate until ready to serve.
Preheat the oven to 350°F. Melt the butter in a large skillet over medium heat. Add the apples and brown sugar. Sprinkle with the cinnamon, nutmeg, and ginger. Cook until the apples are softened.
Whisk together the lemon juice and cornstarch. Add to the apple mixture and cook until the sauce is thickened. Stir in the vanilla. Set aside and let cool.
Roll out the pie dough into a ¼-inch-thick circle on a lightly floured parchment paper-lined baking sheet. Spoon the apple mixture into the center leaving a 2- to 2½-inch border.
Fold the dough over the filling. Whisk together the egg yolk and water and brush the edges with the egg wash. Sprinkle with turbinado sugar.
Bake the crostata for 45 minutes or until the edges are golden brown and the crust is cooked through. Serve warm with whipped cream or vanilla ice cream, if desired.