Preheat oven to 325°F
Grease an 8-inch square or round pan. Line the pan with parchment.
Beat butter and 1 cup sugar together until light, fluffy, and the sugar has dissolved.
Add eggs one at a time, beating well after each addition.
Add the vanilla and beat for 3 minutes.
Combine baking powder, flour, salt, nutmeg and cardamom.
Add the dry ingredients to the butter and egg mixture.
Mix the batter until thoroughly combined.
Spoon half of the batter into the prepared pan, smoothing along the bottom with the back of a spoon.
Peel, core, and slice the two Granny Smith apples.
Toss the apples with the cinnamon and 1 teaspoon of sugar, and then arrange slices over the first layer of batter.
Spoon the remaining batter over the apples and smooth it across them.
Sprinkle the remaining teaspoon of sugar over the top of the batter.
Peel, core, and slice the sweet apple.
Arrange the slices over the batter in a decorative pattern, and baste apple slices with melted butter.
Place the cake in the oven and bake for approximately 1 hour 15 minutes to 1 hour and 30 minutes.
Allow to cool until warm.