In a large mixing bowl, combine the flour, sugar, and salt. Add cold, small pieces of butter and use a pastry cutter or your fingers to work the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces.
Gradually add ice-cold water, a tablespoon at a time, mixing gently until the dough just comes together when pressed. Shape it into a flat disk, wrap in plastic, and refrigerate for at least 30 minutes to relax the gluten and chill the butter.
While the dough chills, slice the peaches and place them in a bowl. Add lemon juice, vanilla, and a tablespoon of sugar, tossing gently to coat. Let sit for 15-20 minutes to draw out juices and soften the fruit.
Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper. Remove the dough from the fridge and let it sit for a few minutes. On a lightly floured surface, roll the dough into a roughly 12-inch circle, about 1/8 inch thick.
Transfer the rolled dough to the prepared baking sheet. Spoon the peach filling into the center, leaving about 2 inches around the edges. Fold the edges over the filling, pleating as needed, to create a rustic border.
Brush the exposed crust with beaten egg mixed with a splash of cream or milk. This will give the crust a shiny, golden finish. Optionally, sprinkle a little sugar over the crust for extra sweetness and crunch.
Bake the galette in the preheated oven for 40-45 minutes, or until the crust is golden brown and crisp, and the fruit juices are bubbling and caramelized around the edges. Keep an eye on it to prevent over-browning.
Remove the galette from the oven and let it cool slightly on a wire rack. The crust should be flaky and tender, with juicy, caramelized fruit inside. Serve warm or at room temperature, ideally with a scoop of vanilla ice cream or a dollop of whipped cream.
