Whisk together your flour and sugar in a medium saucepan over medium heat. Cook for about 2 minutes to toast the flour. Be careful not to burn it!
Slowly add in your milk, whisk to combine, and bring your heat to medium-high. Bring the mixture to a simmer, whisk continuously until it is thick and pudding-like. Once it's simmering, cook for one minute more.
(Optional) If your mixture has any lumps, place a colander on top of a heat-proof bowl and push your mixture through and into the bowl. Transfer the mixture into a heat-proof bowl. Cover the mixture with plastic wrap so that it is touching the surface without any air bubbles in between, this will prevent a skin from forming on the top of the mixture. Let cool in the refrigerator for a few hours or overnight. You can also transfer the mixture to a sheet pan, spread it out, cover it with plastic, and put it in the freezer for about 30 minutes to cool faster. Make sure the mixture is completely cool or it will melt your butter.
Add your softened butter to the bowl of your stand mixer and whisk on high until very light and fluffy, about 2-3 minutes.
Once the butter is fluffy, place the cooled flour mixture into a piping bag and slowly pipe it into your butter as your whip. Incorporating slowly ensures a smooth buttercream.
Add in your vanilla and salt, then mix until everything is creamy. Use it to frost your cooled cake. This buttercream will set quickly, so it's best used immediately.
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Notes
Toast the flour and sugar dry in the pan for 2 minutes before adding the milk. This removes the raw flour taste from the finished frosting.
Whisk the milk in slowly and stir constantly to prevent lumps. If lumps form, strain the roux through a fine mesh sieve before chilling.
Press plastic wrap directly onto the surface of the roux before refrigerating. This prevents a skin from forming on top.
The roux must be completely cold before adding it to the butter. Even slightly warm will melt the butter and break the frosting.
Make sure your butter is fully softened to room temperature before whipping. Cold butter will not whip up light and fluffy.
Add the roux to the butter slowly using a piping bag with the mixer running. Adding it all at once can cause the frosting to curdle.
Use ermine frosting immediately after making it. It sets as it sits and loses its light, creamy texture within a few hours.
Do not use ermine frosting under fondant. It is too soft to support the weight and will compress and bulge.
To make chocolate ermine frosting, whip in ¼ cup of sifted cocoa powder at the end.
The roux can be made up to one week ahead and stored refrigerated. Whip fresh into butter the day you need it.