Preheat your oven to 350°F (175°C) and position a rack in the center. Butter and flour a 9-inch (23cm) springform pan, or line with parchment paper.
Core and slice the apples into thin, uniform wedges (approximately 1/8-inch thick). In a small bowl, mix together the sugar, cinnamon, and nutmeg, then gently toss with the apple slices to coat evenly. Set aside.
In a large mixing bowl, cream together the butter and sugar until pale and fluffy – approximately 3-4 minutes if using an electric mixer, or 5-6 minutes by hand.
Add eggs one at a time, beating well after each addition. If the mixture appears to curdle, add a tablespoon of the measured flour to re-emulsify. Stir in the vanilla extract.
In a separate bowl, whisk together the flour, baking powder, cardamom, and salt. Gradually fold into the wet ingredients using a rubber spatula, mixing just until combined.
Transfer the thick batter to your prepared pan, smoothing the top with a spatula. Arrange the apple slices on top in an overlapping circular pattern, starting from the outside edge and working toward the center. Gently press the apples slightly into the batter.
Bake for 40-45 minutes, or until a toothpick inserted into the cake portion comes out clean and the top is golden brown. If the cake is browning too quickly, loosely cover with aluminum foil after 30 minutes.
Allow the cake to cool in the pan for 15 minutes before removing the springform sides. Serve warm or at room temperature, dusted with powdered sugar if desired.