Preheat the oven to 350°F.
Place sliced apples in a large bowl. Add the 1 tablespoon flour and 1 tablespoon cinnamon and toss to coat all the apples.
Cover the bottom of the pastry with a layer of apples, shingling the slices so there are no gaps.
Pour about ¼ cup of the caramel sauce evenly over the apple slices. Repeat the layers, until the pie is slightly overfilled and domed on the top.
Top the apples with remaining butter pieces.
Brush your bottom pie crust edges with reserved egg white.
Place the other pie crust over the top of the filled pie pan and press down and crimp the edges.
Cut slits in the top of the pie so steam can escape.
Cover the pie with aluminum foil and bake for about 25 minutes. After 25 minutes, remove from oven and brush with more of your egg white.
Combine your remaining sugar and cinnamon and sprinkle over the top.
Return to oven and continue to bake uncovered for another 25 minutes, until the pie is golden.
Let rest at room temperature for at least 1 hour to set up.
Serve with ice cream and drizzle with any remaining caramel sauce.