Preheat the oven to 350 degrees F. Lightly grease and flour the bottom and sides of a half sheet pan (13x18-inches).
In a medium bowl, whisk together the flour, baking powder, salt and baking soda. In a large bowl, use a stand mixer or an electric hand mixer to beat the butter and oil on medium speed until smooth. Beat in the sugar on medium-high until light and fluffy (2 to 3 minutes).
Add the eggs and 3 of the egg yolks, one at a time, beating until blended after each addition. Beat in the vanilla and the remaining egg yolk. Add ⅓ of the flour mixture and mix just until blended in. Add half of the buttermilk and beat until blended. Add another ⅓ of the flour mixture and beat on low speed until blended. Beat in the remaining buttermilk. And then beat in the remaining flour mixture.
Scrape the batter evenly into the prepared pan. Bake until a toothpick inserted into the center comes out clean- 20 to 22 minutes. Let the cake cool in the pan.
ADD THE FROSTING:
In a large bowl, use an electric mixer to beat the butter and cream cheese at medium speed until creamy. Gradually add the powdered sugar and cocoa powder, beating at low speed until blended. Increase the speed to medium, and slowly add in the milk and vanilla, beating until smooth.
Microwave the chocolate in a microwave-safe bowl at MEDIUM 1½ to 2 minutes or until melted and smooth, stirring at 30 second intervals. Gradually add melted chocolate to mixture; beat until blended and smooth. If the frosting is too soft for spreading, cover and refrigerate for a few minutes until it firms up a little bit. Otherwise, spread the frosting on top of the cooled cake. Cut into squares, and serve.
Notes
The sheet cake recipe shared here is from Abigail Dodge' Sheet Cake cookbook. The frosting recipe is mine.