Begin by preheating your oven to 375 degrees Fahrenheit.

Carefully slice off the top half inch of each of the four apples. Utilize a melon baller or a spoon to excavate the insides of the apples, taking care to leave about 1/4 inch of flesh next to the skin. Discard the cores and seeds.

Coarsely chop the scooped apple flesh and place it in a medium-sized bowl. Ensure you have at least 1 1/2 cups of chopped fruit. If necessary, peel, core, and dice an additional apple, adding it to the bowl. Drizzle the lemon juice over the fruit and gently toss to coat. Incorporate the dark brown sugar, granulated sugar, flour, cinnamon, nutmeg, and salt into the mixture, stirring until well combined. Distribute the filling evenly among the hollowed-out apples.

Roll out the pie crust and, using a pastry wheel, cut it into strips approximately 1/4 inch thick. Lightly brush the top edge of each apple with the beaten egg. Weave the strips into a lattice pattern atop each apple, using the egg wash to secure them in place. Then, lightly brush the lattice top with more egg wash.

Arrange the apples in an 8-by-8-inch baking dish, adding about 1 inch of water to the base of the dish. Cover the dish with foil and bake for 20 minutes. After this initial baking period, remove the foil and continue baking until the crusts achieve a golden-brown color and the filling begins to bubble, which should take an additional 25 minutes. Once baked, allow the apples to cool to room temperature on a cooling rack before serving.