In a small saucepan, whisk together the milk and flour. Bring to a boil over medium heat, stirring constantly. Continue to cook until thick. (To test for proper thickness, allow a blob of the mixture to drop back into the saucepan. You should see the dollop(s) for about 3 - 5 seconds before it melts flat back into the mixture.) Transfer to a bowl to cool; set aside.

Beat the butter, sugar, and vanilla in the bowl of your electric mixer fitted with the paddle attachment (whisk attachment if that's all you have). Beat on medium-high speed until light and fluffy, about 5 minutes. Scrape the sides and bottom of the bowl as needed.

Switch to the whisk attachment. Add one-fourth of the flour-milk mixture at a time, scraping the bowl down as needed after mixing thoroughly after each addition, a total of about 5 minutes. Taste the frosting in a few different areas. If granules of sugar remain, continue to beat another 2 - 5 minutes until they dissolve. (Using superfine sugar speeds up this process.)

If not using immediately, add to a piping bag or other sealed container and refrigerate until ready to use. Then, allow to reach room temperature or you can pipe it easily, about 25 minutes.