For the chocolate frosting:
To make the cake:
Preheat oven to 350°F. Grease two 8 or 9-inch cake pans with nonstick baking spray or butter and flour, and line with parchment paper.
In the bowl of a stand mixer fitted with a paddle attachment, combine the cake flour, sugar, baking powder, salt and baking soda. Add the butter, vegetable oil, and 1 cup of the buttermilk. Mix on low speed until moistened, then increase the speed to medium high and beat until light and creamy, about 4 minutes total. Stop and scrape the bowl down at least once.
Whisk together the eggs, egg yolks, remaining ¼ cup of the buttermilk, and the vanilla extract in a bowl or liquid measuring cup.
Add the egg mixture to the mixer in 3 additions, mixing only until incorporated and scraping the bowl down between each addition.
Divide the batter between the prepared pans, about 1lb 8 oz/680 grams of batter in each pan. Bake for 32-35 minutes, or until tops are golden and slightly domed. When gently pressed with a finger, the tops of the cake should bounce back. Let cool in the pan for 5 minutes, then turn out onto a wire rack to cool completely.
To make the frosting:
You can melt the chocolate in double boiler or in the microwave in 30 second bursts. Heat just until a few bits of chocolate remain, then whisk until smooth. Cool mostly to room temperature before using.
Add powdered sugar, butter, cooled melted chocolate, cream/half and half, vanilla and salt in the bowl of an electric mixer and beat on medium high speed until combined and super creamy. You'll need to scrape down the bowl a few times to make sure it's smooth and uniform. If needed, add cream/half and half a little at a time to get the consistency you want.
To assemble the cake, place one layer (flat side up) on a plate or cake stand. Cover with 1-2 cups of the chocolate frosting. Top with second cake layer and cover the top and sides of the cake with the frosting.
Serve immediately or as desired, though the cake will be its best within 2 days of assembly.
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Prep cake layers in advance and store wrapped in plastic wrap in the fridge for up to 2 days and in the freezer (double wrapped) for up to 3 months. Assemble cake layers while cold from the fridge, and thaw frozen cakes in the fridge overnight still in their plastic wrap before using. See my tutorial on how to store cake layers for more detailed info.
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Prep the frosting in advance and store at room temperature for several days, in the fridge for 1-2 weeks, or in the freezer for several months. For best results, beat frosting until smooth before using, adding a bit more cream if needed to get the right texture.
Calories: 650kcal, Carbohydrates: 80g, Protein: 7g, Fat: 36g, Saturated Fat: 22g, Polyunsaturated Fat: 2g, Monounsaturated Fat: 10g, Trans Fat: 1g, Cholesterol: 146mg, Sodium: 440mg, Potassium: 228mg, Fiber: 2g, Sugar: 59g, Vitamin A: 1006IU, Vitamin C: 0.03mg, Calcium: 84mg, Iron: 2mg