Preparation

Make the cake

• Position a rack in the lower third of the oven and heat the oven to 325°F. Have ready an ungreased, not nonstick 10×4-inch (16-cup) angel food cake pan.
• Separate the eggs, putting the yolks into a small bowl and the whites into a medium bowl, adding the extra white.
• In a large bowl, stir the cocoa powder and 5 Tbs. hot tap water until blended; it will be thick. In another large bowl, stir together the coffee liqueur or espresso and espresso powder.
• Sift the flour into the bowl of a stand mixer or a large mixing bowl. Whisk in 1-1/4 cups of the granulated sugar, the baking powder, and salt. Add the egg yolks, sour cream, oil, and vanilla. Using the stand mixer fitted with the paddle attachment or a hand-held mixer, beat on medium-low speed until blended. Increase the speed to medium high and beat until lighter in color and thick enough to form a slowly dissolving ribbon when the paddle or beaters are lifted, about 3 minutes.
• Add half of the batter to the cocoa mixture. Gently fold in the chopped chocolate until blended.
• Stir the espresso mixture, add the remaining batter to it, and fold until blended.
• Using the stand mixer fitted with a clean bowl and the whisk attachment or a hand-held mixer, beat the egg whites on medium-low speed until foamy, about 1 minute. Increase the speed to medium, and beat until the whites are opaque and form very soft peaks, about 1-1/2 minutes. Slowly add the remaining granulated sugar, and beat until the whites are thick and shiny and form soft peaks, about 1-1/2 minutes.
• Evenly divide the whites between the chocolate and espresso batters and gently fold each until just blended with no visible streaks of either whites or batter.
• Pour about one-third of the chocolate batter evenly around the bottom of the cake pan and top with about the same amount of the espresso batter. Repeat the layering, alternating between the remaining batters. Each layer will not completely cover the previous one.
• Bake until deep golden brown and springy when touched, about 60 minutes. The top will have cracks around the middle that will look moist inside. Immediately invert the pan onto its feet or by sliding the center tube onto the neck of a bottle, if necessary. Let cool completely. Remove from the pan and place on a flat serving plate.

Make the glaze

• Combine the milk or dark chocolate, heavy cream, and coffee liqueur or espresso in a heat-safe bowl. Set over a saucepan of barely simmering water and heat, stirring, until the chocolate melts and the mixture is smooth. (Or microwave the chocolate at 50% power for 1 minute; stop and stir. Microwave it in 15-second increments, stopping and stirring, until it is melted.)
• Set aside to cool, stirring frequently, until thick but still pourable, about 15 minutes. If necessary, add more cream in 1 Tbs. increments to thin.
• Scrape the glaze over the top of the cake. Using an offset spatula, spread it over the top, nudging it gently over the outside edge and allowing it to drip down the side of the cake.
• The cake can be served immediately or covered and stored at room temperature for up to 3 days.