Step 1: In a medium saucepan, whisk together the milk and flour until smooth. Heat over medium heat, whisking constantly, until the mixture thickens considerably and begins to bubble. It should be very thick, like a paste. This will take about 5-7 minutes. Make sure to cook out any raw flour taste.
Step 2: Remove the flour-milk paste from the heat and stir in 1/2 cup of the granulated sugar until dissolved. Transfer the paste to a shallow bowl or plate, cover the surface directly with plastic wrap to prevent a skin from forming, and let it cool completely to room temperature. This can take 1-2 hours on the counter or about 30-45 minutes in the refrigerator (stirring occasionally). It must be completely cool.
Step 3: In a large mixing bowl, using an electric mixer with the paddle attachment (or a hand mixer), beat the softened butter on medium-high speed until light and fluffy, about 2-3 minutes.
Step 4: Add the remaining 1/2 cup of granulated sugar to the butter and continue beating for another 2-3 minutes until well combined and light.
Step 5: With the mixer on low speed, gradually add the completely cooled flour-milk paste, a spoonful at a time, to the butter mixture. Continue to beat on medium-high speed for 5-10 minutes. The mixture may look curdled or separated at first, but keep beating. It will eventually come together and become incredibly light and fluffy.
Step 6: Stir in the vanilla extract and pinch of salt. Beat for another minute until fully incorporated. Use immediately or store for later.