Melt 1 tablespoon of butter in a large skillet 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. Off the heat, pour it into the apple mixture. Stir to thicken. Cool completely—the filling shouldn’t be warm when you add it to the pie shell!
Prepare the crust:
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. It will be dry and seem like it's not going to happen, but keep compressing small amounts in your fists. Do not add additional water! When you have a bunch of small lumps of dough, combine them into two roughly equal discs. (Don't worry about overworking the dough here. Since you've added so little water, it's not as easy for the crust to go tough.)
At this point, you can actually roll out the dough right away (no need to chill!) If you're making the dough in advance, you can place the discs in airtight reusable silicone bags (or wrap in plastic) and chill in the fridge or freeze them.
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.
If you've chilled your dough, 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.