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. Stir in the vanilla and coffee liqueur. Remove the bowl from the pan; set aside until cool and thick but still pourable, about 1 hour.

Meanwhile, make the filling: Put the milk chocolate in a microwave-safe bowl; microwave in 30-second intervals, stirring, until smooth. Let cool.

Beat the butter and salt in a large bowl with a mixer on medium speed until smooth and fluffy, 2 to 3 minutes. Add the melted milk chocolate and beat until combined. Gradually beat in the confectioners' sugar until smooth. Add the cocoa powder and vanilla, increase the mixer speed to medium high and beat until smooth, 2 to 3 minutes.

Assemble the cake: Put 1 chocolate cake layer on a platter; spread 3/4 cup filling on top. Top with a vanilla cake layer and spread with another 3/4 cup filling. Repeat to make 4 layers, ending with a vanilla cake layer. Spread the remaining filling on top and refrigerate until firm, at least 30 minutes. Pour the ganache over the cake, letting it drip down the sides. Let set at room temperature before slicing, about 30 minutes.

Photograph by Levi Brown