In a mixing bowl, combine the all-purpose 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 pea-sized pieces.
Sprinkle ice water, one tablespoon at a time, over the mixture, gently mixing with a fork or your hands after each addition. When the dough begins to come together, press it into a rough ball, being careful not to overmix. Wrap in plastic and chill in the fridge for at least 30 minutes.
While the dough chills, prepare the peach slices by washing, pitting, and slicing them thin. Toss the slices with sugar, cinnamon, and vanilla extract if using, then set aside to macerate slightly.
Preheat your oven to 400°F (200°C). Remove the dough from the fridge and let it sit for a few minutes to relax. On a lightly floured surface, roll the dough into a rough 12-inch circle, about 1/8-inch thick.
Transfer the rolled dough to a parchment-lined baking sheet. Arrange the sliced peaches in the center, leaving about a 2-inch border around the edges. Fold the edges of the dough over the fruit, pleating slightly to create a rustic border, and pinch to seal.
Brush the crust with the beaten egg to give it a shiny, golden finish. Place the galette in the oven and bake for about 40 minutes, or until the crust turns deep golden and the filling is bubbling at the edges.
Once baked, remove the galette from the oven and let it cool for a few minutes. The crust should be flaky and crisp, with a fragrant, caramelized glaze, and the peaches beneath will be juicy and bubbling.
Slice and serve the peach galette warm or at room temperature, enjoying the contrast of flaky crust and tender, syrupy fruit with each bite.
