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Place the chocolate in a microwave-safe bowl and heat at 50% power in 15-second intervals, stirring after each interval, until chocolate is melted and smooth. Set aside to cool for 5 minutes.
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Place the butter in a medium mixing bowl and beat with an electric mixer until very smooth. Add 5 tablespoons of milk, the vanilla, and salt and beat to combine.
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With mixer on low, gradually beat in 5 cups of powdered sugar. Mix in the white chocolate.
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Turn mixer to medium and beat until frosting is smooth. Add add additional powdered sugar if frosting is too thin; if it’s too thick add additional milk.
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Transfer about 3/4 of a cup to another bowl (this is your filling).
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Add the espresso powder to the frosting that remains in the first bowl. Pour a teaspoon of milk on top of the powder (to help the powder dissolve) and beat on high until the espresso powder is incorporated.
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Keep beating until most of the espresso powder is mostly dissolved so your frosting isn’t gritty.
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If your frosting is thin after adding the extra milk, add a little more powdered sugar to thicken it. Transfer about 1/2 cup of frosting to a piping bag fitted with a large round tip.
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Place 1 cake layer on a cake stand or cardboard cake circle. Pipe a circle of frosting around the edge of the cake. Spread the filling inside the circle.
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Top with a second cake layer, then frost the cake. Decorate with white chocolate curls, or as desired. Serve.
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Uneaten cappuccino cake can be stored at room temperature, covered, for 2 to 3 days or in the fridge, covered, for 3 or 4 days. (I think it tastes best at room temp so I recommend setting refrigerated slices on the counter for a bit to warm up!)
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You can also freeze individual slices, tightly wrapped, for up to 2 months.