FOR THE CAKE: Adjust oven rack to middle position; heat oven to 350°F; grease two 9-inch-round by 2-inch-high cake pans with softened butter; dust pans with flour and knock out excess.
Combine chocolate, cocoa powder, and hot water in medium heatproof bowl; set bowl over saucepan containing 1 inch of simmering water and stir with rubber spatula until chocolate is melted, about 2 minutes. Add 1/2 cup sugar to chocolate mixture and stir until thick and glossy, 1 to 2 minutes. Remove bowl from heat and set aside to cool.
Whisk flour, baking soda, and salt in medium bowl. Combine buttermilk and vanilla in small bowl. In bowl of standing mixer fitted with whisk attachment, whisk eggs and yolks on medium-low speed until combined, about 10 seconds. Add remaining 1 1/4 cups sugar, increase speed to high, and whisk until fluffy and lightened in color, 2 to 3 minutes. Replace whisk with paddle attachment. Add cooled chocolate mixture to egg/sugar mixture and mix on medium speed until thoroughly incorporated, 30 to 45 seconds, pausing to scrape down sides of bowl with rubber spatula as needed. Add softened butter one tablespoon at a time, mixing about ten seconds after each addition. Add about one-third of flour mixture followed by half of buttermilk mixture, mixing until incorporated after each addition (about 15 seconds). Repeat using half of remaining flour mixture and all of remaining buttermilk mixture (batter may appear separated). Scrape down sides of bowl and add remaining flour mixture; mix at medium-low speed until batter is thoroughly combined, about 15 seconds. Remove bowl from mixer and fold batter once or twice with rubber spatula to incorporate any remaining flour. Divide batter evenly between prepared cake pans; smooth batter to edges of pan with spatula.
Bake cakes until toothpick inserted into center comes out with a few crumbs attached, 25 to 30 minutes. Cool cakes in pans 15 minutes, then invert onto wire rack. Cool cakes to room temperature before frosting, 45 to 60 minutes.
TO MAKE FROSTING: Melt chocolate in heatproof bowl set over saucepan containing 1 inch of barely simmering water, stirring occasionally until smooth. Remove from heat and set aside. Meanwhile, heat butter in small saucepan over medium-low heat until melted. Increase heat to medium; add sugar, corn syrup, vanilla, and salt and stir with heatproof rubber spatula until sugar is dissolved, 4 to 5 minutes. Add melted chocolate, butter mixture, and cream to clean bowl of standing mixer and stir thoroughly to combine.
Place mixer bowl over ice bath and stir mixture constantly with rubber spatula until frosting is thick and just beginning to harden against sides of bowl, 1 to 2 minutes (frosting should be 70 degrees). Place bowl on standing mixer fitted with paddle attachment and beat on medium-high speed until frosting is light and fluffy, 1 to 2 minutes. Stir with rubber spatula until completely smooth.
TO FROST CAKE: Place one cake layer on serving platter or cardboard round. Spread 1 1/2 cups frosting evenly across top of cake with spatula. Place second cake layer on top, then spread remaining frosting evenly over top and sides of cake. Cut into slices and serve.