Preheat the oven to 350°F / 175°C. Grease a 13 × 18in / 33 × 46cm half-sheet pan, line it with parchment paper, and grease the paper.
Melt 1⅓ cups of semisweet chocolate, 4 tablespoons of butter, ¾ cup of vegetable oil, ¾ cup of water, and half of 1 cup of unsweetened cocoa powder together, stirring until smooth.
Whisk 2 cups of all-purpose flour, 2 cups of sugar, ½ teaspoon of baking soda, and ½ teaspoon of salt in a large bowl, then mix in the chocolate mixture.
Fold in ¼ cup of buttermilk, 2 large eggs, 2 large egg yolks, and 2 teaspoons of vanilla extract until just combined. Spread evenly in the pan.
Bake for 18-20 minutes, until a cake tester comes out clean. Cool completely in the pan.
For the icing, melt 2 cups of semisweet chocolate with 2 tablespoons of light corn syrup, 1 teaspoon of vanilla extract, and ⅔ cup of heavy cream, stirring until smooth. Beat in 1 cup of icing sugar and the remaining cocoa.
Spread the icing over the cooled cake and leave to set for 30-60 minutes.
