Special equipment: a 10-inch pie pan
For the dough: In a large bowl, mix the flour, granulated sugar and salt. Using a pastry blender or your fingers, cut in the cold butter until pea-size.
In a separate bowl, beat 4 of the eggs with a fork until fluffy and pale yellow.
Using a fork, mix the eggs into the flour mixture until the dough just comes together. Finish combining the dough with your hands to form a ball, being careful to avoid over-working the dough.
Cut the ball in half, forming two smaller balls of dough. Gently press down on each to form two equal disks. Wrap each disk in plastic wrap and refrigerate for at least 2 hours or up to overnight.
For the filling: In a bowl, coat the apple slices with the lemon juice. Add the brown sugar, granulated sugar, flour, cinnamon and nutmeg, and toss to evenly coat. Set aside.
Preheat the oven to 375 degrees F.
Dust a clean surface with flour. Using a rolling pin, roll out the dough, one disk at a time, to 1/4-inch-thick, about 12 inches in diameter.
Spray the pie pan with nonstick cooking spray. Roll one portion of the dough onto the rolling pin and carefully lift into the pan, leaving the excess dough hanging over the edge. Spoon the fruit filling into the pan, creating a mountain of fruit. Then push down the center of the filling pile to even it out, and trim the crust. Roll the second portion of the dough onto the rolling pin and place over the filling, with the excess dough hanging around the edges of the pan. Flute the edges of the crust by pinching dough between your thumb and forefinger all the way around the edge. Cut holes in the top crust with a paring knife or poke holes with a fork to allow steam to escape. Beat the remaining egg, and brush the piecrust with the egg wash using a pastry brush or your fingers. Sprinkle the crust with the turbinado sugar. Bake until golden brown and the juices begin to bubble, about 1 hour.
Cool the pie for several hours before cutting and serving (if you can wait that long!).