1. Make the pie crust: In a large bowl, whisk together 2 cups all-purpose flour and 1 tsp salt. Cut in 1 cup cold, cubed unsalted butter until the mixture resembles coarse crumbs with some pea-sized pieces. Gradually add 0.5 cup ice water, mixing gently until the dough just comes together. Be careful not to overmix. Divide the dough in half, flatten each half into a disk, wrap in plastic, and chill for at least 30 minutes.

2. Prepare the filling: In a large bowl, combine the 6 peeled, cored, and sliced apples with 0.5 cup granulated sugar, 0.25 cup brown sugar, 0.02 cup all-purpose flour, 1 tsp cinnamon, 0.25 tsp nutmeg, and 1 tbsp lemon juice. Toss gently to coat the apples evenly.

3. Assemble the pie: On a lightly floured surface, roll out one dough disk into a 12-inch circle. Carefully transfer it to a 9-inch pie dish. Pour the apple filling into the crust.

4. Roll out the second dough disk. You can place it over the filling, cutting several slits in the top for steam to escape, or create a lattice top for a decorative finish. Trim any excess dough from the edges and crimp the edges to seal.

5. For a golden crust, brush the top crust with the beaten egg wash and sprinkle lightly with a tablespoon of sugar.

6. Bake: Preheat your oven to 425°F (220°C). Place the pie on a baking sheet to catch any drips and bake for 15 minutes. Then, reduce the oven temperature to 375°F (190°C) and continue baking for another 45-50 minutes, or until the crust is golden brown and the apples are tender when pierced with a knife. If the crust edges start to brown too quickly, you can cover them loosely with aluminum foil.

7. Cool: Allow the pie to cool on a wire rack for at least 2-3 hours before slicing and serving. This resting period is crucial for the filling to set properly, preventing it from being too runny.