Mix flour, sugar, and salt in a large bowl. Using a pastry cutter or food processor, cut in cold vegan butter until mixture resembles coarse crumbs with some pea-sized pieces remaining.
Combine ice water with apple cider vinegar, then add to the flour mixture 1 tablespoon at a time, mixing just until the dough holds together when pinched.
Divide the dough into two equal portions and form into discs. Wrap each and refrigerate for at least 30 minutes (or up to 2 days).
Peel, core, and slice apples into ¼-inch slices. In a large bowl, combine apples, both sugars, cornstarch, spices, lemon juice, and vanilla. Toss until evenly coated and let sit for 15-20 minutes.
Preheat your oven to 425°F (220°C). Remove one disc of dough from the refrigerator and let stand for 5 minutes to slightly soften.
On a lightly floured surface, roll the dough into a circle about 12 inches in diameter and ⅛-inch thick. Carefully transfer to a 9-inch pie dish.
Pour the apple filling into the prepared crust, mounding slightly in the center. Dot the filling with small pieces of the 2 tablespoons of vegan butter.
Roll out the second disc of dough and either cover the pie completely (cutting vents in the top) or create a lattice pattern.
Trim excess dough, then fold and crimp the edges to seal. Brush the top with plant-based milk and sprinkle with coarse sugar.
Place the pie on a baking sheet and bake at 425°F for 20 minutes, then reduce temperature to 375°F (190°C) and bake for 30-35 minutes more, until golden brown and filling is bubbling.
Allow the pie to cool on a wire rack for at least 3 hours before slicing to let the filling set properly.