Place 4 cups of dried apples and 2 2/3 cups of water in a large saucepan. Bring the mixture to a boil, then reduce the heat to medium-low and simmer for 30 minutes, or until the apples become very soft. Lightly mash the apples and stir in 1/2 cup of white sugar. Set the mixture aside to cool.

Preheat your oven to 400 degrees F (200 degrees C).

In a large bowl, cream together 1/2 cup of shortening and 1 1/4 cups of white sugar until the mixture is light and fluffy. Add 5/8 cup of buttermilk, 3/4 teaspoon of baking soda, 1 tablespoon of baking powder, 1/4 teaspoon of salt, and 1 3/4 teaspoons of ground ginger, stirring to combine. Gradually mix in 3 3/4 cups of all-purpose flour, about 1 cup at a time, until a stiff dough forms. Divide the dough into 5 equal portions. On greased cookie sheets, pat each portion into a 9-inch circle.

Bake the dough circles in the preheated oven for 6 to 8 minutes, or until the edges turn golden. Carefully transfer the baked layers to a cooling rack.

On a serving plate, stack the cake layers, spreading approximately 3/4 cup of the cooled apple filling between each layer. Spread the remaining apple filling over the top layer, then garnish with 1/2 cup of dried apple slices. Allow the assembled cake to stand overnight before serving.