Place a rack in the middle of your oven and preheat to 400°F. Moisten your countertop with a sprinkle of water, place a sheet of parchment paper on it, dust it with flour, and place the pie dough on top. Roll the dough into a circle, about 1/4 inch thick, and use a fork to poke a few holes in the center. Transfer the pie dough on the parchment paper to a sheet pan and chill it in the freezer.
Cut the peaches with their skin on into 1/2 inch thick slices. Place the slices in a large bowl and drizzle with vanilla. Mix the sugar with the cornstarch, then sprinkle the sugar mixture over the peaches.
Toss the peaches gently to cover them in the sugar mixture. Place the peaches in a mesh sieve and place it over the large bowl. Macerate the peaches for about fifteen minutes until they release their juices.
Take the juices (not the peaches) and simmer them in a small pot. Do not stir, as this will crystallize the caramel. Occasionally pick up the pan to swirl the juices. Once the juices thicken and turn a golden brown, remove them from the heat.
Remove the chilled pie dough from the freezer. Layer the peach slices on the dough, working from the center out. Leave a four-inch border. Top the peaches with the cooled caramel.
Fold the outer lip of the dough over the peaches, pleating it where necessary. You’re not looking for perfection here; that’s the beauty of a galette. Lightly brush a thin layer of heavy cream onto the top of the galette.
Bake for 40 minutes, or until the crust is crispy and golden. Cool the galette on the sheet pan for fifteen minutes before serving. Use the edges of the parchment paper to help you lift it off the sheet pan.