Preheat oven to 350 degrees F with rack in the middle position. Generously butter three 8-inch round cake pans. (I only have 2 pans so I bake 2 layers first and then bake the 3rd layer.) Cut out rounds of parchment paper to fit the bottoms of the pans, place parchment rounds in the pans. Butter the parchment and then dust pans with flour, tapping out the excess.

In a medium bowl, whisk together the flour, cocoa powder, instant espresso powder, baking soda, and salt.

Place the milk and vanilla in a liquid measuring cup.

In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium-high speed until smooth and shiny, about 30 seconds. With the mixer running, slowly add the sugar and beat until the mixture is pale, light, and fluffy, 3 to 5 minutes. Add the eggs 1 at a time, beating for 1 minute after each addition.

With the mixer on low speed, add about a third of the flour mixture, followed by a third of the milk mixture. Mix until barely incorporated and repeat with remaining flour and milk mixtures. Scrape the sides of the bowl and beat on low speed for 15 seconds more.

Divide the batter evenly between the three cake pans, spreading it out into even layers. Bake cakes until a tester comes out clean or with just a few crumbs attached, 23-26 minutes. Let cakes cool in pans on wire racks for 10 minutes.

In a small bowl, melt the chopped chocolate in the microwave. (Heat 1 minute on high power, stir, and then heat in 15 second intervals, stirring in between, until melted.) Set aside to cool slightly.

In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium speed until pale and fluffy, about 6 minutes.

Mix in the melted chocolate, milk, vanilla, and salt and beat until well combined.

With the mixer on low speed, gradually beat in the powdered sugar and cocoa powder and then continue to mix on medium speed until frosting is light and fluffy, about 4 minutes.

Place one of the cake rounds on a cake stand or plate. Spread a layer of frosting on top with a small offset spatula. Carefully place a second cake layer on top. Spread a layer of frosting on top and spread evenly. Set the third cake layer on top. Place a generous sccoop of frosting on top, spreading evenly with a small offset spatula and working your way down the sides until you have an even layer of frosting over the entire cake.

Cake can be stored at room temperature for a few hours; cover and store leftover cake in the refrigerator. Let cake come to room temperature before serving.