Stir together the flour and the ½ teaspoon of the salt in a large bowl. Hold this aside.
Slice the cold butter into small cubes.
Add the cubed cold butter, the ¼ cup of sugar, and the flour mixture to the bowl of a food processor. Pulse the machine to just combinethe ingredients. (You can also a stand mixer.)
Add in the eggs one at a time and pulse the food processor to blend them into the butter-flour mixture, scraping down the bowl once they're both in. Mix in the milk until the batter is lumpy like cottage cheese.
Lightly flour a work surface and turn out the contents of the bowl. Squeeze the dough together and knead just a few times until the dough comes together. (A bench scraper is helpful here.) Do not overwork it or the crust will be too tough.
Split the dough in half, flatten each half with your hand and wrap the dough portions in plastic wrap. Place them in the fridge for an hour.
In the meantime make the filling:
Run about 4 cups of cold water into a large bowl. Stir in the 2 teaspoons of salt until it's dissolved. (This will be for the cut apples to prevent them from turning brown.)
Peel and core each apple, and slice the apples into 1-inch chunks. As you cut the apples place the chunks into the salt water and stir them around so that all sides get wet.
Have a large saucepan ready on the stovetop. When the apples are all cut, drain them in a colander and rinse off all of the saltwater. Place the apples in the saucepan.
Add in the 1 cup of sugar and 1 teaspoon of the vanilla extract. Heat the contents over medium heat until the apples begin to simmer.
Stir and simmer the apples for about 8 minutes until the apples have softened but still have a little bite in them. Remove the pan from the heat and allow the apple mixture to cool to room temperature. There will be some liquid: that's okay.
Assemble the pie:
Preheat oven to 375° F. Place a metal baking sheet in the oven (large enough to hold the pie dish.)
Have ready a 9-inch pie plate. I prefer a metal pie plate as they cook the bottom of the pie better. If using a glass or ceramic pie pie plate, move the oven rack down one level.
Take one disk of dough from the fridge. On a lightly floured surface roll out the dough to about a 12-inch circle, ¼-inch thick. (This dough tends to be a little sticky; sprinkle flour on the board and rolling pin to help with this.)
Use your rolling pin to help lift the crust into the pie plate. Press the dough into the bottom and up the sides, allowing a 1-inch overhang.
Stir the remaining teaspoon of vanilla extract and the almond extract into the apple mixture. Spoon the apples and liquid into the pie crust and spread them to an even layer.
Roll out the second dough disk on a lightly floured surface to 12 inches in diameter, ¼-inch thick. Lift the dough carefully to the top of the pie. Trim the edge to be just a little longer than the bottom edge, then roll the top edge underneath the bottom edge all around. (If it's too thick in a spot, trim it back a little.) Crimp the edge all around.
Cut a few slashes or cutouts into the top crust to vent the pie. Cover the top loosely with plastic wrap and pop it in the freezer for 5 minutes.
Bake that pie:
Place the pie in the oven on the hot baking sheet. Bake it for about 45-55 minutes, until the top crust is light, golden brown color.
Remove the pie and place it on a cooling rack. Move it to a cool spot in the kitchen or a porch.
Now here's the hard part: let the pie cool completely! If it's cut while warm, the filling will run out.
Dust the top of the pie generously with powdered sugar. Slice the pie and serve it with fresh whipped cream, ice cream, or crème anglaise.
Note: this pie can also be made with the flaky Best-Loved Flaky Pie Crust Recipe. It is less sweet than the crust above and flakier.
Notes
This pie dough is a sweet pâté sucrée which results in a more tender crust that's delicious with this filling. Another great dough is my Best-Loved Flaky Pie Crust recipe: less sweet and flakier. Both crusts are wonderful choices for this flavorful filling.