Make the pie crust: In the work bowl of a food processor, pulse the flour, sugar, and salt together to blend. Add the butter and pulse until the largest chunks of butter are about the size of peas.
Transfer 1⁄4 cup ice-cold water to a measuring cup with a spout, and stir in the vinegar. With the machine running, drizzle this mixture into the food processor. Once the initial liquid has been added, drizzle in another 2–4 tbsp. of ice water as needed, pulsing, until the crust is uniformly moist and begins to gather in the bottom of the work bowl. (The machine’s tone will change.)
Dump the dough into a mixing bowl and press into a ball. Use as directed, or separate into two mounds, press into 1 1⁄2-inch discs, wrap well in plastic, and refrigerate 1 hour before using, or up to 3 days.
Separate the dough into two roughly equal portions. Working with a floured rolling pin on a lightly floured board, roll each portion into a 12-inch round. Line a baking sheet with parchment paper. Lightly flour each round, stack them on the prepared sheet, and refrigerate for 1 hour. (You can also cover and refrigerate for up to 24 hours.)
Make the filling: In a large bowl, toss together the apples, both sugars, flour, cardamom, cinnamon, and sea salt to blend. Fold in the hazelnuts and set aside.
Remove the dough from the fridge. Fit one round into the bottom of a 9-inch pie pan (the deep-dish kind works best here), allowing the excess dough to hang over the sides. (If you’ve chilled the dough overnight, you may need to leave it at room temperature for 5–10 minutes before it’s soft enough to bend into the pan.) Dump the apple mixture into the crust, shaking the apples around as needed to settle them into the pan.
Transfer the second dough round to a large cutting board, and using a large, sharp knife, cut it into 10 roughly 1-inch-thick strips. Brush the interior edge of the fitted bottom crust and the strips lightly with the egg mixture.
Transfer the strips to the top of the pie, egg-side up, placing the longest strips across the center and the shorter strips toward the edges. You can either place five strips in one direction, then simply layer the remaining five strips in the other direction, or you can scour the Internet for instructions on making a lattice-top pie. Gently press the strips into the bottom crust where they meet.
Using scissors or a small, sharp knife, trim the crust to about 1⁄2-inch beyond the edge of the pan. Fold the bottom crust over the strips all the way around the pie, then use your fingers or a fork to crimp the edges closed. Brush the edges with the egg mixture, then transfer the pie to a clean parchment-lined baking sheet and refrigerate for another 30 minutes, or up to 4 hours. Reserve the egg wash.
When ready to bake, preheat the oven to 375°F.
Brush the pie all over with another thin layer of egg wash, and bake 75–90 minutes (longer if the pie’s been chilling for a while), or until the pie is well browned and the filling bubbles slowly anywhere you can see it. Let the pie cool for at least an hour, or overnight, before serving