In the bowl of an electric mixer, beat butter, sugar, and vanilla until pale and creamy. Mix in the flour, cocoa powder, baking powder, baking soda, and salt until combined. Add the milk and mix until combined.
Roll the dough into balls (I like to portion the dough by using a 1 1/2 tablespoon cookie scoop). Place balls of dough 2-3 inches apart on a baking sheet. Gently press dough down with your hand or the bottom of a glass that is dipped in sugar (about 1/3 inch thick).
Bake for 13 minutes. Let cookies cool for 3 mintues on the pan before moving them to a cooling rack to cool completely.
For the Frosting:
Place butter, cocoa powder, and milk in a small sauce pan and cook over medium heat until butter is melted. Remove from heat and whisk in the powdered sugar.
Working one at a time, spoon 2-3 teaspoons frosting onto each cookie, spreading the frosting to the edges and then immediately adding sprinkles. (Frosting sets quickly so it is important to frost and sprinkle one at a time. You'll want to stir the frosting in the pan between applying to the cookies. If it starts to thicken and becomes difficult to spread smoothly onto the cookies just place the pan back on the stove over medium heat for about 15-25 seconds while whisking until it thins again.)