Preheat an oven to 350°F Using a small pastry brush, grease 2 sandwich cookie cake pans with shortening or softened butter, making sure to coat the details and crevices in the pans. Dust thoroughly with cocoa powder; tap out excess.
To make the cakes, put the chocolate and butter in a heatproof bowl and microwave until melted, stirring once, about 1 1/2 minutes. Alternatively, put the chocolate in a heatproof bowl and set over but not touching barely simmering water in a saucepan. Melt the chocolate, stirring occasionally. Remove the bowl from the heat and stir in the butter, a few pieces at a time, until blended. Let the chocolate mixture cool slightly.
In a bowl, whisk together the flour, the 1/2 cup cocoa powder, the baking powder and salt until combined. Set aside.
In a large bowl, whisk together the granulated sugar, eggs and vanilla until well combined. Add the chocolate mixture and whisk until incorporated. Whisk in the flour mixture until combined. Divide the batter between the prepared pans and spread evenly. Gently tap the pans on the counter to release any air bubbles. Bake until the tops of the cakes are shiny and a toothpick inserted into the center comes out with a few crumbs attached, about 35 minutes.
Transfer the pans to a wire rack and let cool for 15 minutes, then invert the cakes onto a cutting board.
Meanwhile, make the buttercream frosting: In the bowl of an electric mixer fitted with the flat beater, combine the confectioners' sugar, butter, the 3 1/2 Tbs. milk, the vanilla and salt and beat on low speed until combined, about 1 minute. Stop the mixer and scrape down the sides of the bowl. Increase the speed to medium and beat until fluffy, about 3 minutes. If the frosting is dry, add more milk, 1 teaspoons at a time, until it is creamy but still holds peaks.
Using a large spatula, carefully turn 1 cake over so the printed side faces down. Spread the buttercream evenly over the flat side of the cake. Place the second cake, printed side up, on top. Cut into slices and serve. Serves 12 to 16.