Mix the flour and a pinch of salt in a bowl, then add the cold butter and use your fingers or a pastry cutter to work it into coarse crumbs, resembling a sandy texture. This keeps the dough tender and flaky.
Gradually pour in the ice water, a tablespoon at a time, gently mixing until the dough begins to come together. Handle it minimally to avoid toughness.
Form the dough into a rough disk, wrap it in plastic wrap, and refrigerate for at least 30 minutes to relax the gluten and keep it cold.
Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper for easy cleanup.
On a lightly floured surface, roll out the chilled dough into a rough circle about 12 inches in diameter. Keep the dough cool and handle it lightly to prevent tearing.
Transfer the rolled dough onto the prepared baking sheet. Spread a thin layer of almond paste in the center, leaving about 2 inches border around the edges. This adds richness and helps prevent sogginess.
In a bowl, toss the peach slices with lemon juice, granulated sugar, and brown sugar until well coated. Layer the fruit over the almond paste, leaving the border clear for folding.
Fold the edges of the dough over the peaches, pleating slightly as you go, to create a rustic crust. Press down gently to seal and encourage even baking.
Mix the beaten egg with milk and brush this wash over the crust to promote a golden, crisp finish. Sprinkle a little sugar on top if desired.
Bake the galette in the preheated oven for about 40 minutes, or until the crust is deep golden brown and the fruit juices are bubbling over the edges.
Remove from the oven and let cool slightly on the baking sheet before serving. The crust should be flaky and crisp, with warm, juicy peaches peeking through.
