Cook milk and flour together in a saucepan over medium-high heat, stirring constantly, until thickened to the consistency of pudding. Bring to a boil, then pour into a heat-proof container. Lay a piece of parchment paper directly on the surface to prevent a skin from forming. Refrigerate for 8 hours to overnight.

Cream sugar, butter, and salt in a stand mixer or using a hand mixer until sugar granules disappear.

Remove boiled milk mixture from the refrigerator and stir with a spatula to ensure there are no lumps. Slowly pour into creamed butter mixture and mix on low speed until completely incorporated. Stop the mixer and scrape down the bowl. Increase the speed to medium and beat until frosting is light and fluffy, 5 to 10 minutes. Beat in vanilla extract.