Preheat oven to 425 degrees. Cover a large baking sheet with parchment paper.
Whisk the flour, sugar, and salt together in a medium bowl. Using a pastry knife, cut in the butter. Stir in 4-6 tablespoons of water. Start with 4 tablespoons and add ½-1 tablespoon of water at a time just until the dough pulls together.
On a lightly floured surface, pat and shape the dough into a disk and cover tightly with plastic wrap. Chill in the refrigerator for at least 1 hour up to overnight.
Gently roll the dough into a 12-13-inch circle on a lightly floured surface. Use the rolling pin to carefully move it to the prepared baking sheet. If you have space in the refrigerator, slide the baking sheet in while you mix the peach mixture.
Mix the peaches, granulated sugar, cornstarch, ground cinnamon, and vanilla extract in a large bowl.
Using a slotted spoon, scoop the peach mixture into the middle of the pie crust. Reserve the juice in the bowl. Pull up 2-3 inches of the dough at a time over the fruit, overlapping the dough with each fold. Gently pinch overlaps in the dough. Drizzle any juice left in the bowl over the peaches. Brush the beaten egg over the pie crust edges. If using, sprinkle the piecrust with coarse sugar.
Bake for 25-30 minutes until the fruit is bubbly and the crust is golden brown.
Notes
Store the galette covered in the refrigerator for up to 4 days. If possible, store it in an airtight container.
Reheat in the oven on a baking sheet at 375 degrees for 10 minutes.