In a large mixing bowl, combine 2 ½ cups flour and ½ cup sugar. Cut in the chilled butter until the mixture resembles coarse crumbs.

Beat the egg and vanilla extract together, then mix into the flour mixture until combined. Form the dough into two balls, wrap in plastic, and refrigerate for at least one hour.

In another bowl, toss the sliced apples with ¾ cup sugar, 2 tablespoons flour, lemon juice, cinnamon, and nutmeg. Allow the mixture to sit and release juices.

Preheat your oven to 425°F (220°C).

On a floured surface, roll out one dough ball into a circle. Place it in a pie pan, leaving some overhang.

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

Roll out the second dough ball and place over the apple filling. Trim and seal the edges. Cut slices in the top crust for steam to escape. Brush with beaten egg.

Bake in preheated oven for 15 minutes, then lower the temperature to 350°F (175°C) and bake for 30-35 minutes until golden brown and apples are soft.