Step 1

Use a metal baking pan for this recipe; a glass baking dish will cause the edges of the cake to overbake as it cools. It's important to use bleached cake flour here; substituting unbleached cake flour or a combination of all-purpose flour and cornstarch will cause the cake to fall. To ensure the proper texture, weigh the flour. Our favorite bittersweet chocolate is Ghirardelli 60% Cacao Bittersweet Chocolate Premium Baking Bar.

Step 2

FOR THE CAKE:  Adjust oven rack to middle position and heat oven to 350 degrees. Grease and flour 13 by 9-inch baking pan. Combine eggs and yolks, buttermilk, and vanilla in 2-cup liquid measuring cup and whisk with fork until smooth.

Step 3

Combine flour, sugar, baking powder, baking soda, and salt in bowl of stand mixer fitted with paddle. Mix on low speed until combined, about 20 seconds. Add butter and oil and mix on low speed until combined, about 30 seconds. Increase speed to medium and beat until lightened, about 1 minute. Reduce speed to low and, with mixer running, slowly add egg mixture. When mixture is fully incorporated, stop mixer and scrape down bowl and paddle thoroughly. Beat on medium-high speed until batter is pale, smooth, and thick, about 3 minutes. Transfer batter to prepared pan and smooth top. Tap pan firmly on counter 5 times to release any large air bubbles.

Step 4

Bake until toothpick inserted in center comes out with few crumbs attached, 28 to 32 minutes. Let cake cool completely in pan on wire rack, about 2 hours.

Step 5

FOR THE FROSTING:  Combine sugar, cocoa, butter, hot water, and salt in bowl of stand mixer fitted with whisk attachment. Mix on low speed until combined, about 20 seconds. Increase speed to medium and continue to mix until smooth, about 1 minute longer, scraping down bowl as necessary. Add melted chocolate and whip on low speed until incorporated. Let sit at room temperature until thickened to spreadable consistency, 30 to 40 minutes.

Step 6

Frost cake. Refrigerate until frosting is set, about 20 minutes, before serving.