Gather the cake ingredients.
Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment. Alternatively, use an electric hand mixer. Let stand for about 30 minutes to bring to room temperature.
Meanwhile, sift 1 cup of granulated sugar, cake flour, and cocoa into a medium bowl. Sift again and set aside (reserve the sifter, you will use it again). Position a rack in the center of the oven and heat to 350 F.
With the mixer on medium speed, beat the egg whites until frothy.
Add the cream of tartar and beat on high speed, gradually adding the remaining 1/2 cup of sugar. Continue beating for 3 to 5 minutes, until stiff peaks form. Beat in 1/2 teaspoon of salt and 2 teaspoons of vanilla extract.
Sift half of the flour-cocoa mixture over the beaten egg whites and gently fold it in with a spatula until blended but still streaky.
Sift the remaining flour-cocoa mixture over the egg whites and continue folding until well combined.
Spoon the batter into a two-piece 9.5-inch tube pan. Run a butter knife or spatula through the batter to release any air bubbles.
Bake the cake for about 38 to 42 minutes, or until a wooden skewer or cake tester inserted into the cake comes out clean.
Invert the pan and cool the cake completely.
Slide a knife or spatula around the pan and center tube to loosen the cake. Grasp the center tube to remove the cake from the outer pan; use a knife or spatula to release the cake from the bottom of the pan.
Place the cake on a cake plate or cake stand and dust with confectioners' sugar, or frost with the optional fluffy chocolate frosting. If desired, garnish with shaved chocolate.
Gather the frosting ingredients.
Put the semisweet and milk chocolate chips in a heat-safe bowl; set aside.
Pour the cream into a small saucepan and cook over medium heat until steaming hot. Do not let the cream boil.
Pour the cream over the chocolate chips and let stand for 5 minutes. Stir until melted. If there are still lumps of chocolate, place the bowl over a saucepan of simmering water and stir until smooth. Alternatively, microwave on low in short bursts, stirring frequently, until the mixture is smooth.
Let the chocolate mixture stand, stirring every 15 minutes, until cooled to room temperature, about 70 F to 75 F. Cooling will take about 1 to 2 hours.
Add a pinch of salt, the vanilla, and 2 to 3 pieces of butter to the chocolate mixture. Beat with an electric mixer with whisk attachment on low speed, adding the butter a few pieces at a time until smooth and well combined. Alternatively, use a stand mixer fitted with the whisk attachment.
Increase the speed to high and beat for about 4 to 5 minutes, scraping down the bowl as needed.
Frost the cake, garnish with the shaved chocolate, if desired, and served.