This recipe makes enough to frost 24+ cupcakes the way you see here. For a big cake, a larger batch of cupcakes or a higher frosting to cupcake ratio, you can 1 1/2x or double this recipe. A double recipe requires a larger stand mixer or a very big bowl -- I have the 6 quart bowl lift KitchenAid and it works fine. I recommend sifting in batches. :)
In the bowl of a stand mixer fitted with the whisk (or in a large bowl with electric beaters), beat the butter for three to four minutes until light and creamy. Sift together the powdered sugar and cocoa powder, then add to mixer (with mixer off) along with milk and vanilla. Beat on low until incorporated, then on high for three to four minutes until smooth and fluffy. Let sit for 10 minutes to help powdered sugar chillax a bit into the mix. (Technical term.) If frosting is too thick, beat in up to two tablespoons more milk a little at a time.
This frosting pipes beautifully and spreads equally well with a spatula. I prefer to use it right away, but you can keep it covered in the fridge for several days, bring to cool room temperature and whisk again for a minute to use leftovers.