Prepare the dough: In a large bowl, combine the flour, sugar, and salt. Add the cold butter cubes and use a pastry cutter or your fingers to incorporate the butter into the flour until the mixture resembles coarse crumbs. Slowly add the ice water, a tablespoon at a time, until the dough starts to come together. Form the dough into a disk, wrap it in plastic wrap, and refrigerate for at least 30 minutes.

1. Make the filling: While the dough chills, peel, core, and slice the apples. In a large bowl, toss the apple slices with lemon juice, granulated sugar, brown sugar, cinnamon, nutmeg, flour, and a pinch of salt. Set aside.
2. Assemble the galette: Preheat your oven to 375°F (190°C). On a lightly floured surface, roll the chilled dough into a rough 12-inch circle. Transfer the dough to a parchment-lined baking sheet. Pile the spiced apple mixture into the center of the dough, leaving about 2 inches of border around the edges. Fold the edges of the dough over the apples, pleating as you go, to form a rustic edge.
3. Add the topping: Dot the exposed apples with small pieces of butter and brush the edges of the crust with a beaten egg. Sprinkle the crust with a little granulated sugar for added sweetness and shine.
4. Bake: Bake the galette for 40-45 minutes, or until the crust is golden brown and the apples are tender. If you like a crispier crust, you can bake it a few extra minutes. Remove from the oven and let cool slightly.
5. Serve: Slice the galette and serve warm, ideally with a scoop of vanilla ice cream or a drizzle of caramel sauce.