Set a bowl over a pan of simmering water. Melt chocolate. Remove and let cool. Grease and flour 2 baking rounds. Preheat the oven to 350°F.
In the bowl of a standing mixer combine flour, sugar, cocoa powder, espresso powder, baking powder, and baking soda.
At low speed, add egg, oil, milk, and vanilla. Mix until combined. Mix in boiling water. Add batter to two pans. Bake 18-22 minutes. Let cool then flip onto a cooling rack.
Beat butter until fluffy. Add sugar and vanilla. Pour in the chocolate. Add the heavy cream. Frost the cake. Use a spoon to swirl the frosting.