Prepare the crust: Pulse first 5 ingredients in a food processor until mixture resembles coarse meal, 5 to 6 times. While pulsing, gradually add ice-cold water, 1 tablespoon at a time, pulsing just until dough begins to come together. Divide dough in half; shape into 2 flat disks. Wrap each in plastic wrap; chill 2 hours or up to 2 days.

Prepare the filling: Preheat oven to 375°F. Toss together apples, granulated sugar, and 2 tablespoons of the flour in a bowl until well coated. Divide apples between 2 large parchment paper-lined rimmed baking sheets. Bake until just tender, about 30 minutes, rotating baking sheets between top and bottom racks halfway through baking time. Cool 30 minutes. Transfer to a bowl. Fold in brown sugar, cinnamon, nutmeg, salt, and remaining 2 tablespoons flour until combined. Chill 1 hour.

Assemble pie: Unwrap 1 dough disk. Roll out on a lightly floured surface into a 1/4- to 1/8-inch thickness. Fit into a lightly greased (with cooking spray) 9-inch deep-dish pie plate. Repeat rolling procedure with remaining disk. Cut 6 (1 1/4-inch-wide) strips, and set aside. Cut 3 (3/4-inch-wide) strips from remaining dough. Braid thinner strips.

Spoon filling into prepared pie plate, smoothing to form an even surface; dot with butter. Arrange 1 1/4-inch-wide dough strips and the braid in a lattice design over filling. Trim excess dough from edges; fold dough under to create a clean edge. Chill pie 30 minutes.

Preheat oven to 375°F. Whisk together egg and 1 tablespoon water until well blended. Brush entire pie with egg wash. Bake on lowest oven rack until crust is golden and filling is bubbly, 1 hour to 1 hour and 20 minutes, shielding pie with aluminum foil after 50 minutes, if needed. Cool on a wire rack 2 hours before serving.