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TO MAKE THE CAKE: Preheat the oven to 350F.
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Place 1 oven rack in the top third of the oven and the second in the bottom third.
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Line three 9-inch round cake pans with parchment paper rounds, grease with butter or cooking spray, dust the pans with flour, and knock out the excess.
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Melt the 1 cup butter and chocolate in a metal bowl set over a saucepan with 2 inches of lightly simmering water.
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Stir constantly until the chocolate is completely melted.
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Whisk in the granulated sugar and vanilla extract.
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In a small bowl, stir the baking soda into the sour cream.
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Whisk the sour cream-soda mixture into the melted butter and chocolate; whisk in the eggs, 1 at a time, then whisk in the flour until thoroughly combined.
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Add the 2 cups hot coffee and whisk until smooth.
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Spoon the batter evenly into the prepared cake pans.
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(Each cake pan will have about 3 cups of batter.)
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Set 2 filled pans on 1 oven rack and the remaining pan on the other.
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Arrange the cake layers on the racks so that no layer is directly over another.
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Bake until the cake is firm to the touch and a toothpick inserted in the middle comes out clean, 35 to 40 minutes.
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Monitor the layers carefully for doneness; each one may be done at a different time.
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Cool the layers in their pans about 10 minutes; unmold onto wire racks to cool completely.
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TO MAKE THE FROSTING: Dissolve the espresso powder in the 1/3 cup hot brewed coffee.
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Set aside to cool.
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Using an electric mixer fitted with the paddle attachment, beat the 2 pounds butter and powdered sugar on medium-high until light and fluffy, about 3 minutes.
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Add the cooled espresso and vanilla paste and beat on medium-high speed until even lighter and fluffier than before, an additional 4 to 5 minutes.
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TO ASSEMBLE THE CAKE: Stack 1 cake layer on a serving plate and spread the top with about 1 cup of the frosting.
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Repeat with the second cake layer and another 1 cup frosting.
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Stack the final cake layer on top of the first 2 and generously cover the cakes top and sides with the remainder of the frosting.
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Garnish with fresh strawberries, if desired.
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The cake can be made up to 1 day in advance, frosted, and refrigerated.
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Take the cake out of the refrigerator about 3 hours before serving.
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Serve at room temperature.