In a mixing bowl, combine the flour and a pinch of salt, then add the cold, cubed butter. Use your fingertips or a pastry cutter to work the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces.
Pour in the vanilla extract and lemon juice, then add cold water a tablespoon at a time, mixing gently until the dough just comes together. Be careful not to overwork it; the dough should be crumbly but hold when pressed.
Gather the dough into a ball, flatten into a disk, wrap tightly in plastic wrap, and chill in the fridge for at least 1 hour. This helps relax the gluten and keeps the crust flaky.
While the dough chills, prepare the peaches by slicing them into thin wedges and tossing with lemon juice to brighten their flavor and prevent browning. Set aside.
Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper to prevent sticking.
Remove the chilled dough from the fridge and place it on a lightly floured surface. Roll it out into a roughly 12-inch circle, about 1/8 inch thick, with a floured rolling pin. The edges can be uneven for that rustic look.
Transfer the rolled dough onto the prepared baking sheet. Arrange the sliced peaches in the center, leaving about a 2-inch border around the edges. Drizzle with vanilla extract if using.
Fold the edges of the dough over the peaches, pleating as you go to create a freeform crust. The center will remain open, showcasing the fruit.
Brush the crust with the beaten egg using a pastry brush, then sprinkle with coarse sugar for a crunchy, sweet finish.
Bake the galette in the preheated oven until the crust is golden brown and bubbling around the edges, about 40 minutes. The aroma of baked fruit and butter will fill your kitchen.
Remove from the oven and let cool slightly before slicing. The flaky crust will crackle as you cut, revealing the tender, juicy peaches inside. Serve warm or at room temperature and enjoy!
