You can make a flavor swap by substituting the cardamom in the spiced peach filling for 1 teaspoon each of cinnamon and nutmeg.
Prepare the Crust: Place first 3 ingredients in the large bowl of a food processor; pulse 5 times to combine. Add the cubed butter and shortening. Pulse 12 to 15 times until mixture resembles small peas. Evenly sprinkle 8 tablespoons of the ice water over the top of the flour and butter mixture. Pulse 5 times to combine. Pinch dough; if it does not stick together, add up to 2 tablespoons ice water, 1 tablespoon at a time, pulsing 3 to 5 times to combine after each addition. Turn dough out onto a work surface, and knead 4 to 5 times to bring together. Shape into a 1-inch-thick oval. Wrap with plastic wrap, and chill 1 to 24 hours.
Preheat oven to 425°F. Let dough stand at room temperature 15 minutes to slightly soften. Place on a lightly floured sheet of parchment paper, and roll into a 12- x 16-inch oval about 1/8 inch thick. Transfer dough and parchment to a large baking sheet; chill until ready to use.
Prepare the Filling: Stir together the sugar, cardamom, and cornstarch in a large bowl. Cut each peach into 8 wedges, and add to sugar mixture. Gently toss to coat. Add the lemon juice, vanilla, and ginger; gently toss to combine.
Remove peaches from the bowl, reserving 4 tablespoons of the liquid. Arrange the peaches in a single layer on the dough, overlapping slightly, leaving a 1 1/2-inch border along each edge. Cut the pastry border every 4 inches, and fold each piece up over the fruit. Brush crust with heavy cream, and, if desired, sprinkle with turbinado sugar. Drizzle reserved 4 tablespoons liquid from bowl over the peaches.
Bake in preheated oven 15 minutes. Reduce heat to 350°F, and bake until golden brown, about 40 minutes. Cool 10 minutes on baking sheet; transfer galette to a wire rack, and cool 1 hour.