Preparation

1. Preheat the oven to 350°F with the rack in the middle position. Spray a 13x9-inch cake pan with nonstick cooking spray and line the bottom with parchment paper. Reserve.
2. Put the flour, sugar, baking powder, and salt into a large mixing bowl. Using the beaters, mix on Speed 1 to fully combine, at least 30 seconds. Reserve.
3. Put the buttermilk, eggs, yolks, and vanilla extract into a medium bowl. Mix on Speed 3.
4. Reduce to Speed 2. Gradually add the oil to the wet ingredients, and mix until combined, then increase to Speeds 3 to 4 and mix until smooth, about 1½ minutes.
5. Gradually add add the liquid mixture to the cake batter, mixing on, mixing on Speed 2 until completely combined.
6. Pour the batter evenly into the prepared cake pan. Tap gently on the countertop to remove any bubbles, and then place in the oven. Bake until the center springs back to the touch and a cake tester comes out clean when inserted in the center, about 25 minutes.
7. Remove from the oven. Put the pan on a wire rack until cool to the touch, then transfer the cake to a wire cooling rack to cool completely before frosting.
8. While cake is cooling, prepare the frosting. Put the sugar, chocolates, butter, and vanilla extract into a large mixing bowl. Using the whisk, mix on Speed 2 for 2 minutes. Add 10 tablespoons of the milk. Increase speed to 4 and beat until smooth and fluffy, scraping down the bowl as needed, about 4 minutes. If the frosting seems thick or stiff, add milk, 1 tablespoon at a time, until easily spreadable.