In a large mixing bowl or stand mixer fitted with paddle attachment, mix to thoroughly combine the cream cheese and butter. Add the remaining confectioners' sugar and stir until smooth. If the frosting is too thick to spread, add milk, 1 teaspoon at a time, to smooth and soften it.
Add the vanilla and 1/3 of the confectioners' sugar and mix on low speed until combined.
Add the milk, starting with one teaspoon and adding a second teaspoon if needed, until the frosting is thick and spreadable.