Preparation
Place a rack in the center of the oven, and preheat to 325.
1. TO MAKE THE CAKE: place the egg whites and cream of tartar in a medium mixing bowl. Beat with an electric mixer on high speed for 2-3 minutes, until stiff peaks form. Set aside.
2. Place the cake mix, egg yolks, water, oil coffee powder, and vanilla extract in a large mixing bowl, and with the same beaters used to beat the egg whites (no need to clean them), blend with an electric mixer for 1 minute. Stop, and scrape down the sides of the bowl with a rubber spatula. Increase the mixer speed to medium, and beat for 2 minutes longer, scraping the side down if needed. The batter should look well blended. Turn the beaten egg whites out on top of the batter, and with the spatula, fold the whites into the batter until the mixture is light but well combined. Pour the batter into an ungreased 10" tube pan, smoothing the top with the spatula.
4. Bake the cake until it springs back when lightly pressed, and a toothpick inserted in the center comes out clean, 48-52 minutes. Remove the pan from the oven, and immediately turn it upside down over the neck of a glass bottle to cool for an hour. Remove the pan from the bottle. Run a long, sharp knife around the edge of the cake, and invert it onto a rack, then invert it again onto a serving platter.
5. TO MAKE THE FROSTING: Place the butter and cocoa in a medium mixing bowl. Blend with an electric mixer on low speed for 30 seconds. Stop, and add the sugar, milk, and vanilla extract. Increase the mixer speed to medium and beat until fluffy, 1 minute longer. Spread the frosting on top and sides of the cake with clean, smooth strokes. Slice and serve.