Prepare the pie crust: In a large mixing bowl, combine flour, salt, and sugar. Cut in the cold butter using a pastry cutter or your fingertips until the mixture resembles coarse crumbs. Gradually add ice water, 1 tablespoon at a time, until the dough comes together. Divide the dough in half, shape into disks, wrap in plastic, and refrigerate for at least 1 hour. This step ensures a flaky, tender crust.

Prepare the filling: In a large bowl, toss the sliced apples with granulated sugar, brown sugar, cinnamon, nutmeg, allspice (if using), lemon juice, flour, and vanilla extract. Make sure each apple slice is coated evenly with the spices and sugar for a flavorful, well-balanced filling.

Assemble the pie: Preheat the oven to 425°F (220°C). On a floured surface, roll out one disk of dough into a 12-inch circle and place it into a 9-inch pie dish, letting the edges hang over slightly. Pour the apple filling into the crust, spreading it evenly. Roll out the second dough disk and place it over the apples. Trim excess dough and crimp the edges to seal. Cut small slits in the top crust to allow steam to escape.

Brush and prepare for baking: Brush the top crust with the beaten egg and sprinkle a little sugar for a golden, slightly caramelized finish. This will give the pie a beautiful, glossy appearance.

Bake the pie: Place the pie on a baking sheet to catch any drips. Bake at 425°F (220°C) for 20 minutes, then reduce the heat to 350°F (175°C) and bake for an additional 40 minutes, or until the crust is golden brown and the filling is bubbly. If the edges brown too quickly, cover them with foil.

Cool and serve: Let the pie cool for at least 2 hours before slicing so the filling can set properly. Serve slices warm or at room temperature. For an extra treat, pair with vanilla ice cream, whipped cream, or a drizzle of caramel sauce.