Preheat oven to 350°.
Cream together the butter and sugar in the bowl of a stand mixer for a full 3 minutes.
Scrape sides and add in vanilla, eggs and milk. Mix until combined.
Add flour and baking powder to the bowl. Mix slowly at first gradually getting faster until the dough comes together.
Use a cookie scoop to portion out the dough equally. Flatten the tops of the cookies slightly prior to baking.
Bake at 350° for 8-12 minutes. Pull them out of the oven when they are no longer wet and glossy on top. Pull them out before they turn brown to insure a soft, pillowy cookie.
Allow the cookies to cool on the cookie sheet prior to frosting.
To make the frosting, cream the butter until smooth. Slowly add in the powdered sugar one cup at a time alternating with the liquids (vanilla and milk) until all ingredients have been added and the frosting is at the desired consistency. (add more powdered sugar if you'd like it thicker, add a little more milk if you want it thinner.)