In a large bowl, combine flour and salt; cut in cold, cubed butter until the mixture resembles coarse crumbs. Gradually add ice water, mixing until a dough forms; do not overmix.

Divide the dough in half, form into discs, wrap in plastic, and refrigerate for 30 minutes.

In a large bowl, gently toss sliced apples with granulated sugar, ground cinnamon, nutmeg, lemon juice, and all-purpose flour until well combined.

Preheat oven to 425°F (220°C). Roll out one dough disc and carefully place it into a 9-inch pie plate. Trim the edges and spoon the apple filling evenly into the crust.

Roll out the second dough disc and place it over the filling, crimping the edges to seal. Cut slits in the top crust for ventilation, then brush with beaten egg and sprinkle with sugar.

Bake for 15 minutes at 425°F (220°C), then reduce the oven temperature to 375°F (190°C) and continue baking for another 20-25 minutes, or until the crust is golden brown and the filling is bubbly and tender.