Preheat the oven to 375 degrees F.
Cut the top 1/2 inch from the 4 apples. Using a melon baller or spoon, dig out the insides of the apples and reserve, leaving 1/4 inch flesh next to the skin. Remove the cores and seeds and discard.
Coarsely chop the scooped fruit and add it to a medium bowl; you should have at least 1 1/2 cups. If not, peel, core and dice the additional apple and add it to the bowl. Sprinkle the lemon juice over top and toss to coat. Add the brown sugar, granulated sugar, flour, cinnamon, nutmeg and salt to the bowl and mix. Divide the filling among the hollowed-out apples.
Roll out the pie crust and using a pastry wheel, cut it into 1/4-inch-thick strips. Brush a light coating of egg wash along the top edge of each apple. Weave the strips into a lattice atop each apple, using the egg wash to adhere (see Cook's Note). Lightly brush the top of each lattice with the egg wash.
Place the apples in an 8-by-8-inch baking dish. Add 1 inch of water to the bottom of the dish. Cover with foil and bake for 20 minutes. Remove the foil and bake until the crusts are golden brown and the filling is bubbling, an additional 25 minutes. Place the apples on a cooling rack to cool to room temperature before serving.