Heat the oven to 350ºF.
For the bananas: Melt the butter in a well-seasoned 10-inch cast iron skillet. Stir in the brown sugar and salt and cook over medium heat, stirring, until evenly combined, 1 to 2 minutes. Remove from the heat.
Cut the bananas in half crosswise, then the halves lengthwise in thirds. Arrange the slices, cut side down, on top of the caramel.
For the batter: In the bowl of an electric mixer fitted with the paddle attachment, combine the granulated sugar, brown sugar, and soft butter. Beat on high until light and fluffy, 3 to 5 minutes. Reduce the speed to medium and beat in the eggs, one at a time, then the mashed bananas, sour cream, and vanilla until combined, scraping down the sides of the bowl between additions as needed.
In a medium bowl, whisk together the flour, baking soda, baking powder, cardamom, and salt. Slowly mix the dry ingredients into the banana mixture until evenly distributed. Pour the batter into the skillet. Smooth the top evenly.
Place the skillet on a rimmed baking sheet. Bake until lightly browned and a cake tester inserted into the middle of the cake comes out clean, 40 to 45 minutes. Remove from the oven and let cool completely. To invert the cake from the pan, run a butter knife around the edge of the cake, then place the skillet over very low heat to just warm the bottom, about 2 minutes. Remove the pan from the heat, then carefully invert the cake onto a large plate.