Combine the flour, granulated sugar, and fine sea salt in a food processor fitted with a blade attachment. Pulse briefly to blend. Add the cold butter, cut into 1/2-inch cubes, and pulse 8–10 times until the mixture resembles coarse crumbs with pea-sized butter pieces. Gradually drizzle in ice water, 1 tablespoon at a time, pulsing after each addition until the dough just starts to come together.

Shape the dough gently into a disk, lightly flour all sides, wrap it in plastic wrap, and refrigerate for at least 1 hour or up to 3 days.

Preheat the oven to 425°F. Roll the chilled dough on a floured piece of parchment paper into a rough 12-inch circle. Transfer the parchment with the dough to a rimmed baking sheet and refrigerate while preparing the peaches.

In a small bowl, blend together the granulated sugar, all-purpose flour, and ground cinnamon. Place the sliced peaches in a separate bowl, sprinkle with the sugar-flour mixture, drizzle in the vanilla extract, and fold gently with a spatula until evenly coated.

Arrange the peaches in overlapping concentric circles on the dough, leaving a 2-inch border around the edges. Discard any excess liquid to prevent overflow. Scatter the diced butter pieces over the peaches.

Fold the dough edges inward over the fruit, pleating and pressing gently to seal any cracks. Brush the folded crust with beaten egg and sprinkle with coarse sugar for added crunch.

Bake for 25–30 minutes, or until the crust turns golden and the peach filling bubbles thickly. Allow the galette to rest for about 15 minutes before slicing and serving.