1. Make the Pie Crust:

• Combine Dry Ingredients: In a large bowl, whisk together the flour, salt, and 1 tablespoon granulated sugar.
• Cut in Butter: Add the very cold butter cubes to the flour mixture. Toss lightly to coat the butter. Using a pastry blender, two forks, or your fingertips, cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized (and slightly larger) pieces of butter remaining. Work quickly to keep the butter cold.
• Prepare Liquid: Combine the cold water, ice cubes, and apple cider vinegar in a bowl or measuring cup.
• Add Liquid: Add the ice water/vinegar mixture to the flour/butter mixture, 1-2 tablespoons at a time, mixing gently with a fork after each addition. Stop adding liquid when the dough just begins to come together. Do not add too much liquid. The dough should still be slightly shaggy but hold together when squeezed.
• Divide and Chill: Divide the dough in half. Flatten each half into a flat disk. Wrap each disk tightly in plastic wrap. Refrigerate for at least 1 hour (or up to 2 days) before rolling.

2. Prepare the Filling:

• In a large bowl, combine the shredded apple, blackberries, blueberries, lemon juice, lemon zest, granulated sugar, light brown sugar, cornstarch, cinnamon, cardamom, cloves, and salt. Stir gently until evenly combined. Set aside.

3. Assemble the Pie:

• Preheat Oven and Baking Sheet: Place a large, foil-lined baking sheet on an oven rack in the lower third of the oven. Preheat the oven (with the baking sheet inside) to 425°F (220°C).
• Roll Bottom Crust: On a lightly floured surface, roll out one disk of chilled pie dough into a circle about 12 inches in diameter. Carefully transfer the dough to a 9-inch glass pie pan. Gently press the dough into the bottom and up the sides of the pan. Trim any excess dough, leaving about a 1/2-inch overhang.
• Add Filling: Pour the prepared berry filling into the chilled pie shell.
• Roll Top Crust: Roll out the second disk of chilled pie dough into a circle large enough to cover the pie. You can either place the full crust on top or cut it into strips for a lattice top.
• Place Top Crust: Carefully place the top crust (or arrange lattice strips) over the filling. Trim the top crust, leaving about a 1/2-inch overhang. Fold the overhang of the top crust under the overhang of the bottom crust, pressing gently to seal the edges together. Crimp the edges decoratively (e.g., using your fingers or a fork).
• Cut Vents: Cut several slits in the top crust (or ensure openings in the lattice) to allow steam to escape during baking.

4. Egg Wash and Sugar:

• Prepare Egg Wash: In a small bowl, whisk together the egg white and 1 teaspoon of water.
• Apply: Brush the top crust (and edges) lightly with the egg wash. Sprinkle the 2 tablespoons of raw coarse sugar evenly over the egg-washed crust.

5. Bake the Pie:

• Initial Bake: Carefully place the assembled pie onto the preheated baking sheet in the oven. Bake at 425°F (220°C) for 25 minutes.
• Reduce Temperature and Continue Baking: Lower the oven temperature to 375°F (190°C). Continue baking for another 35 to 40 minutes, or until the crust is deeply golden brown and the filling is bubbling thickly through the vents. If the crust edges start to brown too quickly, cover them loosely with strips of aluminum foil.
• Cool Completely: Remove the pie from the oven. Place it on a wire rack and let it cool completely for at least 2-3 hours (preferably longer) before slicing and serving. This cooling time is essential for the filling to set properly.