Make the dough: In a large bowl, whisk together flour, sugar, and salt. Add the cold butter and use a pastry cutter or your fingers to cut it in until the mixture resembles coarse crumbs. Add ice water, 1 tablespoon at a time, mixing just until the dough holds together. Shape into a disk, wrap in plastic wrap, and chill for at least 30 minutes.

Prepare the filling: In a bowl, gently toss peach slices with sugar, cornstarch, lemon juice, and vanilla (if using). Let sit while the dough chills.

Roll out the dough: Preheat oven to 400°F (200°C). On a lightly floured surface (or parchment paper), roll the chilled dough into a rough 12-inch circle.

Assemble the galette: Transfer dough to a parchment-lined baking sheet. Arrange the peach slices in the center, leaving a 2-inch border. Fold the edges over the fruit, pleating as needed. Brush crust with beaten egg and sprinkle with coarse sugar.

Bake: Bake for 35–40 minutes or until the crust is golden brown and the filling is bubbling. Let cool for 10–15 minutes before slicing.