Four types of chocolate are called for in this decadent dessert: cocoa powder for the cake batter, bittersweet chocolate and milk chocolate for the frosting, and semisweet chocolate for the garnish.
Cake: Preheat oven to 350 degrees. Whisk together cake flour, cocoa, espresso powder, and salt.
In the bowl of an electric mixer fitted with the whisk attachment, beat egg whites on medium speed until frothy. Add cream of tartar and increase speed to high. Continue beating while gradually adding sugar until stiff peaks form, 3 to 4 minutes. Beat in vanilla. Remove bowl from mixer. Add half the flour mixture and gently fold in until streaky, then add remaining flour mixture and fold in until well incorporated (do not overmix, as this will deflate the meringue, creating a dense cake).
Gently spoon batter into an ungreased 9 1/2-inch tube pan with removable bottom; smooth top. Run a knife or small spatula through batter to release air bubbles. Bake until a tester inserted into center comes out clean, about 40 minutes. Invert pan; let cake cool completely in pan. Run a knife around the inner and outer perimeter of the pan to loosen. Use knife to release cake from bottom of pan and remove.
Frosting: Place both chocolates in the bowl of a stand mixer; set aside. In a small saucepan, bring cream and corn syrup to a boil. Pour over chocolate; let stand 2 to 3 minutes.
Whisk, starting in center and working outward, until chocolate is smooth. Let cool at room temperature, whisking occasionally, about 30 minutes.
Attach bowl to stand mixer fitted with the whisk attachment and beat on medium speed, gradually adding butter, until frosting is smooth and silky.
Using a serrated knife, split cake in half horizontally. Place bottom layer on a cake plate. Measure 2 cups frosting and spread evenly with an offset spatula. Top with second layer. Use remaining frosting to cover cake completely, using spatula to create a smooth finish or create swirls. Garnish with chocolate shavings, if desired. Serve immediately.