Heat oven to 350 degrees. Arrange two racks in center of oven. Butter three 8-by-2-inch round cake pans; line bottoms with parchment. Dust bottoms and sides of pans with cocoa powder; tap out any excess. Sift cocoa into a medium bowl, and whisk in boiling water. Set aside to cool.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter on low speed until light and fluffy. Gradually beat in sugar until light and fluffy, 3 to 4 minutes, scraping down sides twice. Beat in vanilla. Drizzle in eggs, a little at a time, beating between each addition until the batter is no longer slick, scraping down the sides twice.
In a large bowl, sift together flour, baking soda, and salt. Whisk milk into reserved cocoa mixture. With mixer on low speed, alternately add flour and cocoa mixtures to the batter, a little of each at a time, starting and ending with flour mixture.
Divide batter evenly among the three prepared pans. Bake until a cake tester inserted into center of each layer comes out clean, 35 to 45 minutes, rotating the pans for even baking. Transfer layers to wire racks; let cool, 15 minutes. Turn out cakes, and return to racks, tops up, until completely cool.
Remove parchment from bottoms of cakes. Reserve the prettiest layer for the top. Place one cake layer on a serving platter; spread 1 1/2 cups chocolate frosting over the top. Add the second cake layer, and spread with another 1 1/2 cups frosting. Top with third cake layer. Cover outside of cake with the remaining 3 cups frosting. Serve.
The cake keeps well for several days stored on the counter: Cover with plastic wrap or a glass dome to keep the cake from drying out.