In a large bowl, whisk together the flour and salt. Cut in the cold, cubed butter using a pastry blender until the mixture resembles coarse crumbs. Gradually drizzle in the ice water, tossing with a fork until the dough just begins to come together.

Divide the dough in half, form each into a flat disk, wrap in plastic, and refrigerate for at least one hour.

In a large bowl, combine the sliced apples, sugars, flour, spices, and lemon juice. Toss to coat.

Preheat oven to 425°F (220°C). Roll out one dough disk and place it in a 9-inch pie plate. Fill with the apple mixture and dot with butter.

Roll out the second dough disk. Create a lattice top or place it whole over the filling, cutting slits for steam. Crimp the edges to seal.

Brush the crust with beaten egg and sprinkle with coarse sugar. Bake on a baking sheet at 425°F for 15 minutes. Reduce heat to 375°F (190°C) and bake for another 35-45 minutes, until golden and bubbly.

Let the pie cool on a wire rack for at least 2 hours before slicing to allow the filling to set.