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1
Preheat oven to 350u00b0F. Grease an 8x8x2-inch square pan with cooking spray or softened butter. Line the bottom with parchment paper.
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2
Combine the flour, cocoa powder, baking powder, baking soda, and salt in a small bowl and whisk long enough to ensure the leavening is evenly distributed.
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3
In a medium bowl, combine the sugars, oil, and vanilla and whisk vigorously. Add the yolk and eggs, one at a time, gently whisking after each addition. Add the buttermilk and whisk again. Dissolve the espresso powder in the boiling water, add this to the bowl, and whisk a final time.
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4
Add the dry ingredients to the wet, and with a rubber spatula, gently fold the dry into the wet, until a few streaks of flour remain. Add the carrots and fold again, until the carrots are completely incorporated.
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5
Transfer the batter to the prepared pan and bake for 40 to 50 minutes, rotating the pan after 15, and checking it with a cake tester/toothpick after 40. The cake is done when the tester comes out with a moist crumb or two.
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6
Remove the cake from the oven and let it cool to room temperature. Alternatively, if you would like to remove the cake from the pan, let it cool for about 10 minutes, until you can comfortably handle the pan, run a paring knife around the edges and invert it right side up onto a serving plate. Let cool completely before frosting.
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7
To make the frosting, combine the butter and cream cheese in the bowl of a stand mixer fitted with the paddle attachment and on medium high speed, beat until smooth and uniform. Add the vanilla and salt and beat again.
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8
Place the confectioners' sugar in a large bowl. With the mixer on low speed, add about a cup of the sugar at a time to the mixer bowl, beating thoroughly after each addition until combined. Once all of the sugar has been added, increase the speed of the mixer to medium or medium high, and beat the mixture until smooth. Do not over mix or the frosting will lose structure.
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9
Generously frost the cooled cake with the frosting, using the back of a large spoon for optimum swirls.
