Core the apples and cut into quarters. Using a mandolin slice the apples into thin strips. It may take a couple tries to get the thickness correct. Apples sliced too thick will break when rolled. Apples sliced too thin will just fall apart. The apple slices should be able to roll up nicely when they are at the right thickness.
Add the apple slices to a medium bowl. Add the cinnamon, sugar and lemon juice. Very carefully mix the apples to coat them. The slices are fragile so it's important to do this step slowly and carefully.
Grease a pie dish with butter. Flour a pastry slab or flat surface and rolling pin. Place down the pie dough and roll out into a large circle, around 1/4 inch thick. Place the rolled out dough into the pie dish. Press down the sides until it is flat all around. Cut off the excess dough using a sharp knife, leaving a 1/2 inch overhang. Fold the dough in 1/2 inch to create the crust. Using your fingers create waves in the crust. (optional)
Add the caramel to the bottom of the crust, reserving around 1/3 cup to drizzle on after. Spread the caramel out with a spatula to coat the bottom. It should be around 1/2 inch thick. If your caramel has thickened, heat it back up over medium low heat to turn it back into a liquid.
Line up 5 of the apple slices on top of eachother, pictured above, on a flat surface. Roll the apple slices up to form a rose. Stick the roses into the caramel. Repeat until the pie is filled with the apple roses.
Drizzle the remaining caramel over the top of the pie. In a small bowl combine the egg and water to create an egg wash. Using a baking brush coat the edges of the pie with the egg wash. This will make the edges crispy and golden brown.
Bake the pie at 375 degrees for 30-35 minutes. Make sure to keep an eye on it. If you notice the apples or crust starting to get to dark, remove from the oven and cover with aluminum foil. Make sure to use kitchen mittens if you do this step to avoid getting burned.
Let the pie set for at least a half hour so the caramel can firm back up. If you can't wait it will be delicious but most likely will fall apart when you cut it. Serve with ice cream or whipped cream and enjoy!