Preheat the oven to 400°F (204°C). Adjust your oven rack to the lower third position.
Add the sliced apples and lemon juice to a large mixing bowl and toss until well combined. Set aside.
In a separate mixing bowl, whisk together the granulated sugar, brown sugar, flour, cinnamon, and nutmeg until well combined. Pour over the sliced apples and mix until the apples are fully coated with the dry ingredients. Set aside for 15 to 20 minutes while you prep the pie crusts.
Roll one of the pie crusts out to about 12 inches in diameter, transfer to a 9-inch pie plate, gently fit it in the dish, and trim any excess dough from the edges. Place in the refrigerator. Roll out the second pie crust dough and either cut lattice strips, decorations or leave whole to fully top the apple pie.
Remove the pie plate from the refrigerator. Using a slotted spoon, scoop the filling into the pie crust (making sure to leave as much juice as possible in the bowl) and spread around into one even layer. Dot with the pieces of butter. Top with the second pie crust and decorate the edges (if topping with a full piece of pie crust, make sure to cut some slits in the top).
Whisk together the egg and milk until well combined. Brush on top of the pie and around the edges. Sprinkle with coarse sugar if desired.
Bake in the lower third of the oven at 400°F (204°C) for 20 minutes, then reduce oven temperature to 375°F (190°C) and bake for an additional 35 to 40 minutes or until the top of the pie is golden brown and the filling is bubbly. Cover the edges of the pie with a pie crust shield if needed to prevent excess browning. You may also cover the top of the pie with aluminum foil to prevent it from browning too much before it's finished baking.
Once the pie is done baking, remove it from the oven, and place it on a wire rack to cool completely. Once cooled, slice and enjoy!