Prepare your double pie crust as per your favorite recipe or using a store-bought alternative. Wrap the dough in plastic wrap and chill it in the refrigerator for at least 30 minutes. Preheat your oven to 425°F (220°C).

In a medium saucepan over medium heat, melt the unsalted butter. Whisk in all-purpose flour, stirring continuously for about one minute. Gradually add water and granulated sugar, bring to a boil, and let simmer for about 3 minutes until slightly thickened.

While the filling simmers, peel, core, and slice the Granny Smith apples. In a large mixing bowl, combine the apple slices with cinnamon, ensuring they are evenly coated. Pour the butter mixture over the apples and toss gently until coated.

On a floured surface, roll out the chilled bottom pie crust to a diameter of 12 inches. Place it into a 9-inch pie plate and spoon the apple filling into the crust, mounding the apples in the center.

Roll out the second pie crust to an 11-inch diameter, cut it into strips, and arrange them over the apple filling in a lattice pattern. Trim any excess dough and pinch the edges to seal. Brush the top crust with the egg wash.

Place the assembled pie on a baking sheet and bake in the preheated oven at 425°F (220°C) for 15 minutes. Lower the temperature to 350°F (175°C) and continue baking for an additional 45 minutes until the filling bubbles and the crust is golden brown.

Once baked, allow the pie to rest at room temperature for at least 1 hour. Serve warm or at room temperature, optionally with vanilla ice cream.