Preheat the oven to 425°F (220°C).

In a large bowl, combine 2 1/2 cups flour, 1 teaspoon salt, and 1 tablespoon sugar.

Cut in 1 cup of chilled, diced butter until the mixture resembles coarse crumbs.

Stir in ice water, one tablespoon at a time, until the mixture forms a ball.

Wrap the dough in plastic wrap and refrigerate for at least 4 hours or overnight.

In another bowl, toss the sliced apples with 3/4 cup sugar, 2 tablespoons flour, cinnamon, nutmeg, and lemon juice.

Roll out half of the dough on a floured surface to fit a 9-inch pie plate. Place the crust in the pie plate.

Fill the crust with the apple mixture and dot with 1 tablespoon butter.

Roll out the remaining dough and place it over the filling. Seal and flute the edges. Cut slits in the top crust to allow steam to escape.

In a small bowl, beat together the egg yolk and milk. Brush the mixture over the crust.

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 on a wire rack before serving.