If you're using this frosting for piping and not for our Monkey Cake, start with Step 2.
In the bowl of an electric mixer fitted with the paddle attachment, combine 1 stick butter, 1/2 teaspoon vanilla, and 1 1/2 cups sugar; mix on medium-high speed until pale and fluffy, about 2 minutes. Stir in food coloring to make pale yellow. Transfer to a small bowl; cover.
Put remaining 3 sticks butter in mixing bowl. Beat on medium-high speed until pale and creamy, about 2 minutes. Reduce speed to medium; add cocoa. Mix until smooth. With mixer running, add remaining 3 3/4 cups sugar, 1 cup at a time. Add remaining 1/2 teaspoon vanilla; mix on high speed until incorporated, 10 to 20 seconds more.