Whisk milk and flour together in a small saucepan over medium-low heat until mixture starts to thicken, 4 to 5 minutes. Then, whisk vigorously until mixture is lump-free, creamy, sticky, and significantly thicker, 2 to 3 more minutes; it may appear to ball up. Immediately remove from the heat and transfer to a plate; let cool to room temperature.

While the milk mixture is cooling, beat butter and sugar in a mixing bowl with an electric mixer for at least 10 minutes. Mixture should be very fluffy; you may need to mix for 5 more minutes. Gradually mix in cooled milk mixture on high speed; continue to mix until fluffy and smooth. Mix in vanilla.

Cover and refrigerate until frosting reaches a spreading consistency, about 30 minutes.