Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper. Lightly flour your work surface and roll out the chilled pie dough into a large circle, about 12 inches in diameter. Transfer the rolled dough onto the prepared baking sheet.
In a bowl, toss the sliced peaches with sugar, lemon juice, and vanilla extract if using. Let them sit for a few minutes until slightly juicy and fragrant.
Spread the peach slices evenly over the center of the dough, leaving about a 2-inch border all around. Dot the filling with small cubes of cold butter to help with caramelization.
Fold the edges of the dough over the filling, pleating as needed, to create a rustic border. The center should remain open, showcasing the fruit.
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 is deep golden brown and the filling is bubbling.
Remove the galette from the oven and let it cool slightly on a wire rack. The crust should be crisp and flaky, with a caramelized, bubbling peach filling.
Slice and serve warm or at room temperature, enjoying the satisfying crunch of the flaky crust along with the sweet, tender peaches inside.
