Melt the chocolate chips in a large microwave-safe bowl for about 3 minutes, stirring every 30 seconds until completely melted. Set aside to cool for 10 minutes.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter until smooth. Add the 1 teaspoon vanilla, ½ cup cocoa powder, and powdered sugar, and beat until well combined. Add the cooled melted chocolate and beat until thick. With the mixer on low speed, beat in 1 tablespoon of milk at a time until the frosting is a smooth, thick, and spreadable consistency.
With a long serrated knife or cake leveler, level the top of each cake layer, if needed, and slice each cake layer in half horizontally. You should have 4 cake layers.
Spread a little frosting on the center of a cake stand or cake plate. Place a cake layer, bottom side up, on the cake stand, and spread a generous layer of frosting on top. Place the next layer, bottom side up, on the frosted layer, and spread with another generous layer of frosting on top. Repeat with the third layer. Place the final layer, bottom side up, and spread the remaining frosting on the top and sides of all the layers.