To make the pastry, combine the flour and salt together in a large bowl. Cut in the chunks of cold butter with a pastry blender, a little at a time, until the dough resembles cornmeal.
Add the 2 egg yolks and the ice water, and blend for a second just to pull the dough together and moisten. Be careful not to overwork the dough. Form the dough into a ball, wrap it tightly in plastic wrap, and let it rest in the refrigerator for 1 hour.
While the dough is resting, prepare the filling.
To make the caramel sauce: place the sugar and water in a small pot and cook, stirring constantly, over medium-low heat until the sugar has melted and caramelized, about 10 minutes. Remove the pot from the burner and add the cream and wine slowly. It may bubble and spit, so be careful. When the sauce has calmed down, return it to the flame, add the vanilla bean and heat it slowly, until the wine and caramel are smooth and continue to slowly cook until reduced by half. Remove from the heat and cool until thickened.
Fill a large bowl with cold water and squeeze in the lemon juice. Peel the apples with a paring knife, cut them in half, and remove the core with a melon baller. Put the apple halves in the lemon-water (this will keep them from going brown). Toss the apples with the flour and cinnamon.
Take the dough out of the refrigerator, unwrap the plastic, and cut the ball in half. Rewrap and return 1 of the balls to the refrigerator, until ready for the top crust. Let the dough rest on the counter for 15 minutes so it will be pliable enough to roll out.
Using a rolling pin, roll out the dough on a lightly floured surface into a 12-inch circle. Carefully roll the dough up onto the pin and lay it inside a 10-inch glass pie pan. Press the dough into the pan so it fits tightly.
Slice a couple of the apples at a time using a mandolin or a very sharp knife. The apples need to be thinly sliced so that as the pie bakes, they collapse on top of each other with no air pockets. This makes a dense, meaty apple pie.
Cover the bottom of the pastry with a layer of apples, shingling the slices so there are no gaps. Ladle about 2 ounces of the cooled red wine caramel sauce evenly over the apple slices. Repeat the layers, until the pie is slightly overfilled and domed on the top; the apples will shrink down as the pie cooks. Top the apples with pieces of the butter.
Now, roll out the other ball of dough just as you did the first. Brush the bottom lip of the pie pastry with a little beaten egg white to form a seal. Place the pastry circle on top of the pie, and using some kitchen scissors, trim off the overhanging excess from around the pie.
Crimp the edges of dough together with your fingers to make a tight seal. Cut slits in the top of the pie so steam can escape while baking. Place the pie on a sheet tray and tent it with a piece of aluminum foil, so the crust does not cook faster than the apples.
Bake the caramel apple pie for 25 minutes on the middle rack. In a small bowl, combine the remaining 1/4 cup of sugar with the freshly grated cinnamon. Remove the foil from the pie and brush the top with the remaining egg white.
Sprinkle evenly with the cinnamon sugar and return to the oven. Continue to bake for another 25 minutes, until the pie is golden and bubbling. Let the apple pie rest at room temperature for at least 1 hour to allow the fruit pectin to gel and set; otherwise the pie will fall apart when you cut into it.