Dissolve 1 tablespoon sugar and 1 teaspoon salt in 2 tablespoons of hot water, and then add an additional 2 ½ tablespoons of cold water. Transfer to the freezer for 20 minutes.
Carefully grate the frozen butter using a box grater or the grating attachment on your food processor. You can also very finely chop it, taking care to minimize contact with your warm hands.
Add 2 1/2 cups all purpose flour to a mixing bowl, along with the grated butter. Combine the butter and flour quickly with your hands for a few seconds.
Drizzle the cold water solution evenly over the flour and butter, and work it with your hands quickly to squeeze it together into 2 balls roughly the same size. It will be a little dry and a tad challenging to ball together, but do your best. Place the dough in airtight reusable silicone bags (or wrap in plastic), and press the balls down into discs.
Transfer the dough discs to the fridge to chill for at least 1 hour (you can also make the dough a day in advance or freeze it if you’d like to make it further in advance).
Make the filling:
While the pie crust is resting, make the filling. Melt 1 tablespoon of butter in a large saute pan over medium high heat. Add the apples, and cook for 3 minutes, stirring to prevent sticking.
Add the cinnamon, lemon juice, and ½ cup of sugar. Cook for 5 minutes, until the apples are somewhat softened. Turn off the heat.
Dissolve the cornstarch in 2 tablespoons of water, and off the heat, pour it into the apple mixture. Stir to thicken. Allow to cool completely—the filling shouldn’t be warm when you add it to the pie shell!
Assemble & bake:
Position a rack in the center of your oven and place a baking sheet on the rack. Preheat the oven to 400°F/200°C.
Remove the dough from the fridge and allow it to sit at room temperature for about 5 minutes to soften slightly, so it’s easier to work with. If one of your dough discs looks larger than the other, take the larger of the two, and roll it out into an 11-12 inch (30 cm) circle. Fold gently in half, and transfer to the pie shell. The dough will break easily, and that’s ok. Patch it and press it back together if needed.
Place the pie shell in the refrigerator while you roll out the top crust. Roll the top crust into a 10-inch circle. Take the pie shell back out of the fridge, and add the cooled filling in an evenly layer. Cover with the top crust, and flute the edges or crimp the edges with a fork.
Brush the crust with egg wash (1 egg beaten with 1 tablespoon of water), and cut vent holes in the top. Place the pie on the heated baking sheet, and bake for 20 minutes. Then rotate the baking sheet 180°, reduce the heat to 375°F/190°C, and bake for an additional 30-35 minutes. Cool for at least 2 hours before slicing and serving.