Step 1

Preheat the oven to 350 degrees.  Grease and flour a 9 inch springform pan.  Make the Apple Topping:  Heat butter in a 12 inch nonstick skillet over medium heat.  Add apples and sauté until they are lightly caramelized, about 5-8 minutes.  Add brown sugar and cinnamon and continue to cook, stirring constantly, until sugar dissolves and apples are evenly coated, about 1 more minute.  Remove apples from the heat and set aside.

Step 2

Make the cake:  Whisk flour, baking powder, salt and cardamom together in a medium bowl and set aside.  Whisk sugars and eggs together in a large bowl until fully combined. Slowly add melted butter to sugar and egg mixture and continue whisking until uniform in texture.  Add sour cream and vanilla and whisk until fully combined. Add flour mixture and whisk until batter just comes together.

Step 3

Scrape batter into prepared pan and even out the top with a small offset spatula.  Arrange apples over the top of the batter in two concentric circles. The apples should just touch each other but not overlap.  Reserve the butter and sugar mixture that remains in the skillet. Transfer the cake to the oven and bake until golden brown and a toothpick inserted into the center of the cake comes out clean or with just a few moist crumbs attached, about 40-50 minutes.

Step 4

Add a couple of teaspoons of water to the skillet along with the reserved butter and sugar mixture leftover from making the apple topping.  Cook over medium heat until warm and syrupy in texture, about 1-2 minutes. Brush syrup over the top of the cake, particularly the apples. Cool cake for at least 20 minutes on a wire rack and serve.