Make the Crust: In a large bowl, combine the flour and salt. Add the chilled, cubed butter and use a pastry cutter or your fingers to work the butter into the flour until the mixture resembles coarse crumbs. Add the ice water, one tablespoon at a time, and mix until the dough comes together. Form the dough into a disk, wrap it in plastic wrap, and refrigerate for at least 30 minutes.

Prepare the Filling: In a large bowl, combine the sliced apples, granulated sugar, brown sugar, ground cinnamon, ground nutmeg, lemon juice, and cornstarch. Toss until the apples are well coated.

Roll Out the Dough: Preheat your oven to 375°F (190°C). On a lightly floured surface, roll out the chilled dough into a 12-inch circle. Transfer the dough to a parchment-lined baking sheet.

Assemble the Galette: Arrange the apple slices in the center of the dough, leaving a 2-inch border around the edges. Dot the apples with the small pieces of butter. Fold the edges of the dough over the apples, pleating as needed to create a rustic look.

Brush and Sprinkle: Brush the beaten egg over the crust and sprinkle with turbinado sugar if desired.

Bake the Galette: Bake in the preheated oven for 40-45 minutes, or until the crust is golden brown and the apples are tender and bubbly.