In a large bowl combine the sliced apples with the sugars, lemon juice, ground cinnamon and salt. Place the apples and their juices in a strainer that is placed over a large bowl (to capture the juices) and let sit at room temperature for 30 mins – 3 hours, or until you have about 1/2 cup (120 ml) of juice.
Pour the collected juice into a 4 cup (960 ml) heatproof measuring cup and add the butter. Place in the microwave and boil the liquid, on high, about 5 to 7 minutes or until the liquid has reduced to about 1/3 cup (80 ml) and is syrupy and lightly caramelized. (Alternatively, you can place the juices and butter in a small saucepan and boil over medium high heat on the stove.)
Meanwhile, remove the top pastry crust from the refrigerator and let it sit at room temperature for about 10 minutes so it has time to soften. Transfer the drained apples slices to a large bowl and mix them with the cornstarch (corn flour). Then pour the reduced syrup over the apples and toss to combine. Pour the apples and their syrup into the chilled pie crust.
Moisten the edges of the pie shell with a little water and then place the top crust over the apples. Tuck any excess pastry under the bottom crust and then crimp the edges using your fingers or a fork. Using a sharp knife, make five- 2-inch (5 cm) slits from the center of the pie out towards the edge of the pie to allow the steam to escape. Lightly brush the beaten egg over the pastry. Cover the pie with plastic wrap and place in the refrigerator to chill the pastry while you preheat the oven.
Preheat the oven to 425 degrees F (220 degrees C). Place the oven rack at the lowest level and place a baking stone or baking sheet on the rack before preheating the oven. Place a piece of aluminium foil on the stone (or pan) to catch any apple juices. Set the pie on the stone or pan and bake for about 45 to 55 minutes or until the juices start to bubble through the slits and the apples feel tender (not mushy) when a toothpick or sharp knife is inserted through one of the slits. Make sure to cover the edges of the pie with a foil ring to prevent over browning after about 30 minutes.
Remove the pie from the oven and place on a wire rack to cool for about 3-4 hours before cutting (during this time the juices thicken and the flavors amalgamate). Serve warm or at room temperature with vanilla ice cream or softly whipped cream.