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1. Preheat the oven to 350u00b0F. Prepare a 10-inch round cake pan by greasing it with cooking spray and then lining the bottom with parchment paper.
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2. To make the cake, pour the hot coffee into a medium bowl and stir in the cocoa powder until it dissolves. Stir in the brown sugar, followed by the yogurt and the vanilla. Stir thoroughly to ensure that all of the ingredients are incorporated.
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3. In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and granulated sugar on medium speed until light-yellow and fluffy, about 3 minutes. Scrape down the sides of the bowl. Add the eggs and mix for 2 minutes, scraping down the bowl as needed.
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4. In a medium bowl, whisk together the flour, baking soda, and salt. With the mixer running on low speed, mix in a third of the flour mixture and half of the coffee mixture. Scrape down the bowl. Add another third of the flour mixture and all the remaining coffee mixture. Remove the bowl from the mixer and, using a rubber spatula, fold in the remaining flour mixture until all of the ingredients are fully incorporated. Pour the batter into the prepared cake pan.
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5. Bake for 25 minutes. Rotate the pan in the oven and bake for 20 more minutes, or until a cake tester inserted in the center of the cake comes out clean. Remove the pan from the oven and let the cake cool in the pan for 20 minutes. Then turn the cake out onto a clean plate, remove the parchment, and turn the cake back over onto a wire rack. Let the cake cool completely.
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6. In a food processor, pulse the brittle pieces 3 to 4 times until the brittle is powdery.
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7. Put the cooled cake on a serving dish. In a microwave-safe dish, heat the dulce de leche on high power for 30 seconds, or until it is just liquid. Spoon the dulce de leche over the cake, and then sprinkle the kosher salt over the dulce de leche. Sprinkle about 1/2 cup of the ground brittle around the outer edge of the cake as a delicious decoration.
