For the cake: Preheat the oven to 350 degrees F. Lightly butter 2 9-inch round cake pans and line the bottoms with parchment.
Whisk the flour, baking soda, baking powder, and salt together in a medium bowl and set aside. Bring 1 1/4 cups water and the milk to a boil in a small saucepan. Remove from heat and keep warm.
Beat the butter in a large bowl with an electric mixer on medium speed until smooth. Add the sugar, increase the speed to medium-high and beat until light and fluffy, stop to scrape down the sides of the bowl occasionally, about 4 minutes. Add the cocoa and vanilla, and beat at medium speed for 1 minute. Reduce the speed to medium-low and add the eggs, one at a time, and beat for 1 minute between each addition. Reduce the mixer to low and beat in the flour in four additions. Carefully pour the hot milk mixture into the batter. Use a rubber spatula to stir until smooth. Evenly divide batter between the prepared pans and lightly tap the pans on the counter.
Bake until the cakes begin to pull away from sides of the pan and the centers spring back to the touch, 30 to 35 minutes. Cool the cakes in the pans on a wire rack for 10 minutes. Invert onto the rack and cool completely.
For the frosting: Bring about 1-inch of water to a simmer in a medium saucepan. Put the milk and semisweet chocolates in a heatproof bowl that sits comfortably in the saucepan without touching the water. Stir occasionally until the chocolate completely melts and is smooth. Let cool slightly and then stir in the sour cream, coffee and vanilla until smooth. Beat with an electric mixer on medium speed until the frosting is silky and fluffy. (Use the frosting right away as it will become firm as it sits.)
Put one cake layer on a serving plate or platter spread about one-third of the frosting all the way to the edge of the cake. Put the second cake layer on top. Frost the top and sides with the remaining frosting.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)