Preheat the oven to 350 °F, and place a baking sheet on the rack below it to catch any drips from the apple pie.

Melt butter in a 10-inch cast-iron skillet over medium heat; add brown sugar, and cook, stirring constantly, 1 to 2 minutes or until sugar is dissolved. Remove from heat, and let the mixture cool while you prepare the apples.

Peel apples, and cut them into 1/2-inch-thick wedges. Toss apples with cinnamon and 3/4 cup granulated sugar.

Place 1 pie crust in the skillet over the brown sugar mixture. Spoon the apple mixture over the bottom pie crust, and top with the remaining pie crust. Slide the edges down along the edges a bit.

Whisk one egg white with a little water until foamy. Brush the top of the pie crust with egg white using a pastry brush; sprinkle with 2 tablespoons of sugar. Cut 4 or 5 slits in top for steam to escape.

Bake for 1 hour to 1 hour and 10 minutes or until golden brown and bubbly, shielding with aluminum foil during the last 10 minutes to prevent excessive browning, if necessary.

Cool on a wire rack for 30 minutes before serving.