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Preheat the oven to 350°F.
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To make the topping: Melt the butter in a 10" cast iron or oven-safe 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.
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Cut the bananas in half crosswise, then cut each piece in thirds lengthwise. Arrange the slices, cut side down, on top of the caramel in the skillet.
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To make the cake: In a large bowl with an electric mixer, beat the sugars and butter at medium-high speed until light and fluffy, about 3 minutes.
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Beat in the eggs one at a time, then the mashed bananas, sour cream, and vanilla, scraping down the sides of the bowl between additions as needed.
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In a medium bowl, whisk together the flour, cardamom, baking soda, baking powder, and salt. Slowly add the dry ingredients to the banana mixture, mixing until thoroughly combined. Scrape the bowl and mix for another 30 seconds.
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Pour the batter into the skillet atop the bananas; smooth the top with a spatula.
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Place the skillet on a rimmed baking sheet and bake until lightly browned and a paring knife inserted into the middle of the cake comes out clean, 45 to 50 minutes. Remove from the oven and let cool completely.
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To serve, run a knife or thin spatula around the edge of the cake to free it from the pan, then place the skillet over very low heat for about 2 minutes to warm the bottom. Remove from the heat, place a serving plate top down over the skillet, and invert the cake onto the plate.
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Let the pan stay on the cake for a few minutes to let any caramel run down before lifting it off to serve.
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Store any leftovers, covered, in the refrigerator for up to five days.
