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Preheat oven to 350°F/175°C. Line a 9-9.5-inch/23-24 cm springform pan with parchment paper.
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Melt the butter and grease the springform with it. Sprinkle pan with 1/2 cup light brown sugar, make sure to get some on the sides of the pan as well.
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Melt butter and stir in the buttermilk. Set aside to cool.
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Put the slices of persimmon in a single layer in the bottom of the pan.
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In a medium bowl, mix flour, baking powder, and salt. Gently stir the butter mixture into the mixture. Add the eggs and vanilla extract and stir until smooth.
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Pour the batter into the prepared pan and smooth the top with a spatula.
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Bake for 35-40 minutes, until cake is golden brown on top and a cake tester comes out clean.
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Let cool for 5 minutes before inverting the cake onto a rack or a plate to cool completely.