Preheat the oven to 425°F (220°C).

In a large bowl, combine sliced apples, granulated sugar, brown sugar, flour, cinnamon, nutmeg, and lemon juice. Toss until apples are well-coated.

Roll out one pie crust and fit it into a 9-inch pie pan.

Pour the apple mixture into the prepared pie crust, spreading it evenly.

Dot the apple filling with diced butter.

Roll out the second pie crust and place it over the filling. Cut slits for venting.

Seal the edges by crimping them with your fingers or a fork.

Brush the top crust with the beaten egg for a golden finish.

Bake in the preheated oven for 15 minutes.

Reduce the temperature to 350°F (175°C) and continue baking for 30-35 minutes until apples are tender and crust is golden brown.