This apple pie is super simple and really lets the flavour of the apples sing. Because of this, make sure to choose apples that are tasty and flavourful on their own.
Preheat oven to 180°C (350°F). Add the butter to a small saucepan and set over medium-high heat. Melt the butter, stirring or swirling occasionally, and cook until it begins to bubble, the bubbles subside and you notice the butter beginning to turn brown on the bottom of the pan. Remove from the heat and set aside.
In a very large mixing bowl, add the sliced apples, flour, brown sugar, lemon zest and juice and browned butter. Mix until very well combined and move to the refrigerator while you roll out your pie crust.
Remove one portion of pie dough from the refrigerator and allow to sit at room temperature for a few minutes. Lightly dust your work surface and both sides of your pastry with flour. Applying even pressure, gently roll the dough into a circle about 30cm in diameter (12 inches). Transfer the dough to a 9-inch pie plate and move it to the freezer for 10 minutes. Repeat with the other dough portion, this time setting on a parchment-lined tray and moving it to the refrigerator.
Remove the pie plate and pastry from the freezer and add all of your apples to the dish. It will mound over the sides of the dish - this is normal. Just ensure that no apple slices are overhanging the edges of your pie plate.
Remove the other pastry round from the refrigerator and gently cover the apples, smoothing it over the apple slices. Use kitchen scissors to trim the excess pastry from the edges and crimp along the edges, either decoratively or with the tines of a fork. Move the pie to the refrigerator for 10 minutes.
Remove the pie from the refrigerator and brush the crust with the beaten egg. Evenly dust the surface with demerara sugar. Cut 3-4 large slits in the pie to vent steam.
Move the pie to a baking sheet and transfer to the oven. Bake the pie for about 1 hour, or until the crust is deeply golden brown and the juices are beginning to ooze from the pastry. Allow the pie to cool completely before serving - at least 4 hours.