In a large bowl, whisk together the flour, sugar, and salt to combine. Add the butter, tossing the cubes in the flour to coat. Working quickly, using a pastry cutter or your fingers, cut the butter into the flour until the butter is about the size of walnut halves.
Make a well in the center and add the water a few tablespoons at a time, and mix just until the dough comes together. Continue adding water a tablespoon at a time until you have a dough that holds together well but isn’t wet or sticky.
Divide dough in half and shape each one into a round disc about 1-inch thick. Wrap both halves in plastic and refrigerate for at least 1 hour.
To Make the Vanilla Chai Apple Filling:
Add apples and apple cider to a large sauté pan set over medium-high heat and cook, stirring occasionally. Meanwhile, whisk together the brown sugar, granulated sugar, cinnamon, ginger, nutmeg, cardamom, allspice, pepper, and flour, and sprinkle this over the pan, stirring to combine. Lower heat slightly and cook until apples have started to soften, approximately 5 to 7 minutes. Remove the pan from the heat, add the vanilla bean paste, and stir until combined. Transfer the apple mixture into a bowl and allow to cool completely. (The apple mixture will cool faster if spread out on a rimmed baking sheet.)
Place a large baking sheet on the middle rack of oven and preheat to 425°F (220°C). Remove the circular disc of dough from the refrigerator and, using a rolling pin, roll it out on a lightly floured surface until it is roughly 12 inches in diameter. Fit this crust into a 9-inch pie plate, trimming it to leave a ½-inch overhang. Place this plate, with the dough, in the fridge.
Remove the other disc of chilled pie dough from the refrigerator and roll the dough out. Using a pastry wheel or sharp knife, cut 16 strips 1/2 inch wide.
Remove the pie plate from the fridge and put the cooled apple pie filling into it. Carefully thread the strips over and under one another, pulling back strips as necessary to weave. Using a pair of scissors, trim the extra overhang. Crimp the edges of the dough with a fork or your fingers. Lightly brush the lattice top with the egg wash and sprinkle with coarse sugar.
Place the pie onto a large baking sheet and bake for 20 minutes. Keeping the pie in the oven, turn the temperature down to 375°F (190°C) and bake for an additional 30-40 minutes, until the filling is bubbling and the crust is deep golden brown. If the top of your pie is getting too brown, cover loosely with aluminum foil.
Let the pie cool on a wire rack to room temperature, at least 3 hours, before slicing, topping with ice cream, and serving.
Store leftovers covered in the refrigerator for up to 3 days.