Preheat the oven to 425°F with a rack in the center position. Melt butter in a large pot over medium heat. Add flour and stir until a paste forms, 1 to 2 minutes. Stir in ½ cup white sugar, brown sugar, ¼ cup water, 1 teaspoon cinnamon, nutmeg, if using, and lemon juice and bring to a boil, stirring often. Reduce the heat to low and simmer until sugars begin to dissolve, 3-5 minutes. Remove the pot from the heat. Add the apples and toss to coat. On a lightly floured surface, roll out the pie dough into two 14-inch rounds. Carefully transfer one round to the pie plate, gently pressing the crust into the dish without stretching. Trim the edges if needed. Spoon the pie filling, including any excess liquid, into the pie crust, forming a slight mound in the center. Place the second pie dough round on top of the apples. Crimp the edges of the bottom crust and top crust together using the tines of a fork or your fingers. In a small bowl combine remaining sugar and remaining cinnamon. Brush the top of the pie with the beaten egg and sprinkle evenly with the cinnamon sugar. Using a sharp knife, cut four, 2-inch slits into the top of the crust. Bake pie on a baking sheet for 15 minutes. Reduce the oven temperature to 350°F and continue to bake, tenting with foil halfway through if the crust is browning too quickly, until the crust is a deep golden brown and the filling is bubbling, about another 45 minutes. Cool on a baking rack at least 2 hours before serving. Rewarm if desired.