Step 1: For the crust, in a large bowl, whisk together 2 ½ cups all-purpose flour, 1 teaspoon salt, and 1 tablespoon granulated sugar.
Step 2: Add the very cold, cubed butter to the flour mixture. Using a pastry blender or your fingertips, cut the butter into the flour until it resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 3: Gradually add ½ cup ice water, one tablespoon at a time, mixing lightly after each addition, until the dough just comes together. Be careful not to overmix.
Step 4: Divide the dough in half, flatten each into a disk, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes.
Step 5: For the filling, in a large bowl, combine the sliced apples, ½ cup granulated sugar, ¼ cup brown sugar, 3 tablespoons flour (or 2 tbsp cornstarch), 1 teaspoon cinnamon, ¼ teaspoon nutmeg, and ⅛ teaspoon allspice. Toss gently to coat.
Step 6: Add 1 tablespoon lemon juice and 1 teaspoon vanilla extract to the apple mixture, tossing again. Let sit for 15-20 minutes.
Step 7: Preheat your oven to 425°F (220°C). Place a baking sheet on the lower rack.
Step 8: On a lightly floured surface, roll out one disk of chilled pie dough into a 12-inch circle. Carefully transfer to a 9-inch pie plate. Trim edges, leaving a 1-inch overhang.
Step 9: Pour the apple filling into the pie crust, mounding it slightly in the center. Dot the top with 2 tablespoons of small butter pieces.
Step 10: Roll out the second disk of dough. Place it over the filling, or cut into a lattice. If using a full top, cut several slits for steam to escape.
Step 11: Trim the edges of the top crust, leaving a 1-inch overhang. Fold the top overhang under the bottom and crimp to seal.
Step 12: Brush the top of the pie with beaten egg wash and sprinkle with 1 tablespoon coarse sugar.
Step 13: Place the pie on the preheated baking sheet. Bake for 15 minutes at 425°F (220°C). Reduce oven temperature to 375°F (190°C) and bake for 40-50 minutes more, or until golden brown and bubbling. Tent edges with foil if browning too quickly.
Step 14: Remove from oven and cool completely on a wire rack for at least 3-4 hours before slicing and serving.