Toss the sliced apples with the granulated sugar, brown sugar, salt, cinnamon, nutmeg, and lemon juice in a large bowl. Let them stand at room temperature for 30 minutes, stirring twice, until the apples have released about 1/2 cup of liquid.

Set a colander over a bowl and drain the apples, saving the liquid. Simmer that liquid in a small saucepan over medium heat for 5 to 7 minutes, until it reduces to about 1/3 cup. Take the pan off the heat, whisk in the cornstarch until smooth, then pour the mixture over the apples and toss to coat. Spread the filling on a rimmed plate to cool completely, about 30 minutes.

Whisk the flour, granulated sugar, and salt in a large bowl. Add the cold cubed vegan butter and cut it in with a pastry cutter or your fingertips until the largest pieces are pea-size and a few flat flakes are still visible.

Stir the apple cider vinegar into the ice water. Drizzle 3 tablespoons over the flour and toss with a fork, adding the rest a tablespoon at a time, until the dough holds together when you squeeze a handful. Gather it into two discs, wrap them, and chill for at least 45 minutes.

Roll one disc into a 12-inch circle on a floured surface and fit it into a 9-inch pie plate, letting the edges hang over. Add the cooled filling and dot it with the tablespoon of vegan butter. Roll the second disc, lay it over the top, trim the overhang, press the edges together, and crimp them. Cut five vents in the top, brush with oat milk, and sprinkle with coarse sugar. Chill the assembled pie for 15 minutes.

Heat the oven to 425 degrees F with a rack in the lower third. Bake the pie for 20 minutes, then reduce the heat to 350 degrees F, cover the rim loosely with foil, and bake for 40 to 45 minutes more, until the juices bubble thickly at the vents and the top is deep golden. Cool on a rack for 3 to 4 hours before slicing.