In the bowl of your electric mixer, fitted with the paddle attachment (or with a hand mixer), beat the butter until smooth. Add the sugar and vanilla extract and beat, on medium high speed, until light and fluffy (about 5 minutes). Add the eggs, one at a time, beating well after each addition. Scrape down the sides and bottom of your bowl as needed. Beat in the cooled cocoa mixture. With the mixer on low speed, add the flour mixture (in two additions) along with the milk (in one addition) and mix only until combined.
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