1. Make the Dough: Place the flour, salt, and sugar into a food processor and pulse twice. Add the cold butter and shortening and pulse about ten times until the mixture resembles coarse crumbs. Sprinkle the cold vodka and water over the dough and pulse 4 to 6 more times until it starts clumping. Turn the dough onto a floured surface, form into a disk, wrap tightly, and chill for at least 45 minutes.

2. Prepare the Apple Filling: While the dough chills, melt the butter in a large sauté pan over medium-high heat. Add the apples, brown sugar, cinnamon, nutmeg, and salt. Cook, stirring often, until the apples have softened but still hold their shape, about 15 minutes. Transfer to a colander to drain and cool for at least 15 minutes.

3. Roll Out the Dough: Let the chilled dough sit at room temperature for 10-15 minutes. Preheat your oven to 425°F (220°C) and position a rack in the lower-middle. Roll the dough into a 20 by 12-inch rectangle and carefully transfer it to a 10×15 inch jelly roll pan, pressing it into the corners.

4. Fill and Seal the Pie: Stir the lemon juice into the cooled apples and spread the filling evenly over the dough. Fold the overhanging dough edges slightly over the filling. Brush the crust with the egg wash and sprinkle with demerara sugar. Cut 6 to 8 small slits in the top for steam to escape.

5. Bake and Cool: Bake for 25 to 30 minutes, or until the crust is a deep golden brown and the filling is bubbling across the surface. Transfer the pie to a wire rack and let it cool completely, for at least 90 minutes, before icing.

6. Add the Maple Icing: In a small bowl, whisk together the sifted confectioners’ sugar, maple syrup, and milk until smooth. Drizzle over the cooled pie. Let it set before slicing and serving.