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Lightly grease a 9-by-13-inch pan with nonstick spray. Heat the oven to 350 degrees.
Make the cake: In the bowl of an electric mixer fitted with the paddle attachment, cream the butter and sugar on medium speed until light and fluffy, 4 to 5 minutes. Beat in the eggs one at a time, then the egg yolk, scraping the sides of the bowl after each addition. Mix on medium speed until fully incorporated, about 1 minute. Add the vanilla extract and mix to combine, 15 to 30 seconds.
In a medium bowl, whisk the flour, baking powder, baking soda and salt to combine. Add about 1/3 of the flour mixture to the mixer and mix on low speed until incorporated, about 30 seconds. Add about half the milk and mix on low speed until incorporated, about 15 seconds more. Repeat with another batch of flour mixture, the remaining milk, then the remaining flour mixture until fully incorporated.
Transfer the batter to the prepared pan and spread in an even layer. Bake until a toothpick inserted into the center comes out clean and the cake is golden and separating from the pan at the edges, 30 to 35 minutes. Cool completely.
Meanwhile, make the frosting: Add the butter to the bowl of an electric mixer fitted with the paddle attachment and beat on medium speed until light and fluffy, 3 to 4 minutes. Remove the bowl from the electric mixer, and sift the confectioners’ sugar and cocoa into the bowl with the butter. Stir with a spatula a few times to incorporate slightly. Return the bowl to the mixer and beat on medium speed until smooth and uniform in color, 1 to 2 minutes. Add the sour cream and melted chocolate and mix on low speed until combined, about 30 seconds. Add the vanilla and salt and mix to combine.
Spread the frosting on the cooled cake in an even layer. Garnish with sprinkles, if desired.