Lattice Crust Apple Pie

Name: Lattice Crust Apple Pie
Ingredients: 2 1/2 cups all-purpose flour; 4 teaspoons sugar; 1/4 teaspoon fine salt; 14 tablespoons cold unsalted butter, diced; 1 large egg, lightly beaten with 2 tablespoons cold water; 2 tablespoons freshly squeezed lemon juice; 3 pounds baking apples like Golden Delicious, Cortland or Mutsu; 2/3 cup sugar, plus more for sprinkling on the pie; 1/2 stick unsalted butter; 1/4 teaspoon ground cinnamon; Generous pinch of ground nutmeg; 1 large egg, lightly beaten