In a medium saucepan off the heat, whisk the flour and sugar together. Slowly stream in the milk while whisking constantly until smooth. Cook over medium heat, whisking constantly, until the mixture thickens to a pudding-like consistency, about 5-8 minutes.

Transfer the pudding to a shallow bowl. Press plastic wrap directly onto the surface to prevent a skin from forming. Let it cool on the counter, then move to the fridge to cool completely to room temperature. This is a critical step.

In a stand mixer with the paddle attachment, beat the softened butter on medium-high speed for 5 minutes until very pale and fluffy.

With the mixer on medium, add the completely cooled pudding to the whipped butter one large spoonful at a time. Once all the pudding is incorporated, add the vanilla and salt. Increase speed to medium-high and beat for another 2-3 minutes until light, airy, and perfectly smooth.