Prepare the Filling: Stir together the sugar, cinnamon, nutmeg, and cornstarch in a large bowl.
Cut each peach into 8 wedges and place in a large bowl. Add sugar mixture to peaches. Gently toss to coat. Add the lemon juice and vanilla; gently toss to combine. Set aside.
Prepare the Puff Pastry: Preheat the oven to 425˚F.
Carefully open one sheet of thawed puff pastry. On a floured surface, roll it out to a 12~inch X 16~inch rectangle. Using a rolling pin, transfer to a half sheet pan.
Remove peaches from the bowl, reserving 4 tablespoons of the liquid. Arrange the peaches in a single layer on the dough, overlapping slightly. Leave a 2~inch border along each edge. Fold each side of the puff pastry up over the fruit. Using a pastry brush, brush the crust with heavy cream, and sprinkle with turbinado sugar. Drizzle reserved 4 tablespoons liquid from the 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.
Serve with a scoop of vanilla ice cream or dollop of whipped cream.