In a large pot, whisk together the brown sugar, cane sugar, apple pie spice, and salt. Add the apple slices and lemon juice and toss to coat. Set aside for 30 minutes to draw some of the juices out of the apples.

In a small bowl, stir together 2 tablespoons of the water and the cornstarch until smooth.

Set the pot with the apples over medium-low heat. Cook, stirring, until the apples soften slightly, about 5 minutes. Raise the heat to medium and pour the cornstarch mixture over the apples. Cook, stirring, until the apple juices bubble and thicken, about 30 seconds. Remove from the heat and stir in the lemon zest and vanilla. Allow the filling to cool completely.

When you're ready to assemble and bake the pie, arrange an oven rack in the bottom third of the oven. Place a baking sheet on the rack and preheat the oven to 425°F with the baking sheet inside it.

Make the bottom crust. Place 1 disk of chilled pie dough on a lightly floured surface. Dust the top with flour and use a rolling pin to roll it out into a large 11- to 12-inch circle. Transfer it to your pie plate and press it firmly into the bottom and up the sides, allowing any excess dough to hang over the edges. Loosely cover with plastic wrap and chill in the refrigerator.

Meanwhile, make the top crust. Roll out the remaining pie dough into an 11- to 12-inch circle.

In a small bowl, whisk together the egg and the remaining 1 tablespoon water. Set aside.

Remove the bottom crust from the refrigerator and pour in the cooled filling, mounding it slightly in the center of the pie. Gently roll the top crust over your rolling pin, then unroll it over the filled pie. Crimp the edges of the pie crust to seal, trimming away any excess dough if necessary.

Brush the top crust (but not the crimped edges) with the egg wash and sprinkle with coarse sugar, if using. Use a sharp knife to cut several steam vents in the crust.

Place the pie on the hot baking sheet in the oven. Bake for 20 minutes, then reduce the heat to 375°F and bake for 45 to 55 minutes, or until the crust is golden brown and any visible juices are bubbling. If you have a glass pie pan, check the color of the sides of the pie. They should also be golden brown. If the crust is getting too brown before 45 minutes, tent it with foil and continue to bake.

Allow to cool completely before slicing and serving, at least 3 hours.

Serve with vanilla ice cream.