In a large mixing bowl, combine the flour, salt, and sugar. Add the cold, cubed butter and use a pastry cutter or your fingertips to cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces.
Gradually add ice-cold water, one tablespoon at a time, mixing gently until the dough begins to come together. You want it to be just moist enough to hold when pressed. Form the dough into a disc, wrap it in plastic, and refrigerate for at least 30 minutes to relax.
While the dough chills, prepare the peaches by peeling, pitting, and slicing them into uneven chunks. Toss the peach slices in a bowl with lemon juice and a tablespoon of sugar, letting the fruit sit and release their fragrant juices.
Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper. Remove the dough from the fridge and let it sit for a few minutes to soften slightly. Roll out the dough on a lightly floured surface into a rough 12-inch circle, handling it gently so it doesn't tear.
Transfer the rolled-out dough onto the prepared baking sheet. Pile the peach slices in the center of the dough, leaving about 2 inches of border around the edges. Fold the edges of the dough over the fruit, pleating as needed to create a rustic, free-form tart.
Mix the beaten egg with milk and brush this egg wash over the crust's surface to give it a shiny, golden finish. This also helps the crust develop a beautiful color as it bakes.
Bake the galette in the preheated oven for about 40-45 minutes, or until the crust is deep golden brown and crisp, and the peaches are bubbling with caramelized edges. Keep an eye on the crust edges; if they brown too quickly, loosely cover with foil.
Remove the galette from the oven and let it cool slightly on a wire rack for about 10 minutes. The crust will crackle as you cut into it, revealing juicy, shimmering peach slices inside.
Slice and serve the peach galette warm or at room temperature, enjoying the contrast of flaky crust and tender, sweet peaches with a drizzle of cream or a dollop of whipped cream if desired.
