Make the apple filling: Combine the dried apples, brown sugar, cinnamon, ginger, nutmeg, and water in a large pot over medium-high heat. Bring to a light boil, then reduce the heat to low. Cook for 30 minutes-1 hour, or until the apples are tender and have absorbed the liquid.
Mash the apples with a potato masher until no large chunks remain. Remove from the heat and let cool.
Make the cake: In a large bowl, combine the flour, baking powder, baking soda, salt, and cinnamon.
In a separate large bowl, cream together the butter and sugar until light and fluffy.
Add the eggs and molasses, and whisk until fully incorporated and the mixture is light brown.
Add ⅓ of the flour mixture to the wet ingredients and fold with a spatula until fully incorporated. Add half of the buttermilk and stir until fully combined. Continue alternating with the remaining flour and buttermilk, fully incorporating each addition before adding the next. The batter will be thicker than a traditional cake batter.
Chill the cake batter for 30 minutes.
Preheat the oven to 350˚F (180˚C). Grease as many 9-inch (22-cm) cake pans as you have and line with parchment paper.
Dump the cake batter onto a floured surface and dust with more flour. Pat into a rectangle, then divide into 5 equal pieces.
Transfer a piece of cake batter to a prepared pan and pat into an even layer. Bake for 12 minutes, until firm. Repeat with the remaining layers. Let cool completely.
To assemble, add a cake layer to a serving plate and top with a layer of apple filling. Spread evenly. Repeat with the remaining cake layers and filling.
Wrap the cake in plastic wrap and chill for 24 hours.
Dust the cake with powdered sugar, then slice and serve.