In a large bowl, mix 2 1/2 cups flour, 1 cup diced butter, 1/4 cup sugar, and 1/2 teaspoon salt until the mixture resembles coarse crumbs.

Stir in ice water, a tablespoon at a time, until the dough comes together. Divide the dough in half, shape into discs, wrap in plastic, and refrigerate for at least 4 hours or overnight.

Preheat the oven to 425°F (220°C).

In a bowl, toss sliced apples with 3/4 cup sugar, 2 tablespoons flour, cinnamon, nutmeg, and lemon juice until well coated.

Roll out one disc of dough on a floured surface to fit a 9-inch pie plate. Place the rolled dough into the pie plate and fill with the apple mixture. Dot with 2 tablespoons butter.

Roll out the second disc of dough and place it over the filling. Seal the edges and cut slits in the top crust for steam to escape.

Brush the top crust with the beaten egg.

Bake in the preheated oven for 15 minutes. Reduce the temperature to 350°F (175°C) and continue baking for 35 to 45 minutes, or until the apples are soft and the crust is golden brown.

Cool the pie on a wire rack before serving.