Place a rack in the middle of your oven and preheat to 400°F. Moisten your countertop and place a 15x15-inch sheet of parchment paper on it. Dust the parchment paper lightly with flour.
Place the pie dough on the parchment paper and roll it out to 14 inches in diameter. Use a fork to poke a few holes in the center of the dough.
Transfer the pie dough on the parchment paper to a sheet pan and chill it in the freezer while you prepare the peaches.
Slice the peaches, skin on, into 1/2 inch thick slices. Place the slices in a large bowl and drizzle with vanilla. Mix the sugar and cornstarch, then sprinkle over the peaches.
Toss the peaches gently to cover them in the sugar mixture, then place them in a mesh sieve set over the large bowl. Macerate the peaches until they release their juices, about fifteen minutes.
Add the juices to a small pot set over medium heat and bring to a simmer. Do not stir. Swirl the juices. Once the juices thicken, remove them from the heat.
Remove the chilled pie dough from the freezer and layer it with the peach slices, 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. 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.