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1
Make the ganache: Heat the heavy cream, semisweet chocolate and espresso powder in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water), stirring, until the chocolate is melted and the ganache is smooth.
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2
Stir in the vanilla and coffee liqueur.
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3
Remove the bowl from the pan; set aside until cool and thick but still pourable, about 1 hour.
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4
Meanwhile, make the filling: Put the milk chocolate in a microwave-safe bowl; microwave in 30-second intervals, stirring, until smooth.
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5
Let cool.
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6
Beat the butter and salt in a large bowl with a mixer on medium speed until smooth and fluffy, 2 to 3 minutes.
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7
Add the melted milk chocolate and beat until combined.
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8
Gradually beat in the confectioners' sugar until smooth.
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9
Add the cocoa powder and vanilla, increase the mixer speed to medium high and beat until smooth, 2 to 3 minutes.
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10
Assemble the cake: Put 1 chocolate cake layer on a platter; spread 3/4 cup filling on top.
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11
Top with a vanilla cake layer and spread with another 3/4 cup filling.
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12
Repeat to make 4 layers, ending with a vanilla cake layer.
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13
Spread the remaining filling on top and refrigerate until firm, at least 30 minutes.
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14
Pour the ganache over the cake, letting it drip down the sides.
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15
Let set at room temperature before slicing, about 30 minutes.
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16
Photograph by Levi Brown
