Place butter, flour, sugar, and salt in the bowl of a mixer with a paddle attachment. On low speed, start mixing. About 1/2 way into mixing, add frozen pork fat and mix until the fat resembles small peas. Add beaten egg with a tbsp of ice water. Work for one second, mix dough with hand to feel moisture. You may need to add more, mix the rest by hand to bring it all together. Wrap dough in plastic and refrigerate for abour four hours or overnight.
Peel and quarter apples, cleaning out the core. On a mandolin, slice the apples paper thin and place in a bowl with all other ingredients and let marinate for at least one hour.
Take 1/3 of the pie dough and roll into a circle about two-inch wider than the pie tin. Place in pie tin so the extra dough is hanging over the pie tin. Place apples in pie tin leaving out the liquid. Pour one cup of the liquid in the pie. Reserve extra liquid for later. Roll out another circle of pie dough but only the circumference of the pie tin. Cut a small circle out from the middle of the dough so that the liquid and steam can release. Place dough on top of apples and roll the overhang around. Refrigerate.
Preheat oven to 375 degrees, brush pie with egg white and sprinkle with a touch of sugar. Bake for about 1 hour, if top of crust starts to get too brown, turn the oven down.
Place 3/4 lb. of sugar and corn syrup in a non-reactive pot on medium heat. Boil until sugar is medium amber, stirring every couple of minutes, meanwhile beat cream until hot. When caramel is medium amber, add reserved sugar in three stages, mixing thoroughly after each addition. Add cream slowly, reducing a little bit each time. Add salt to taste. Let cool.