In a small saucepan, whisk flour into milk and heat, stirring constantly, until it thickens. It will be the consistency of a thick pancake batter or a thick paste. It shouldn't look wet.
Remove from heat and let it cool to room temperature.
Stir in vanilla and desired amount of peppermint extract.
While the mixture is cooling, cream the butter, sugar, and salt together until light and fluffy on medium high, about 5 minutes.
Then add the completely cooled milk mixture. Make sure that the mixture isn't warmed than room temp or it will melt your butter mixture.
Beat it for about 5 minutes on medium-high to high until it looks like whipped cream. It may look separated at first, so keep beating it until it comes together and looks like a whipped cream.
Make sure that your butter isn't too warm! You want it to dent if you press it, but you don't want it to be so warm that it's melting.Also, if you're using a hand mixer, beat it on high. This recipe work well with a stand mixer because it has the power to really whip the frosting.*I like 1/2 teaspoon peppermint and 1/2 teaspoon vanilla for a nicely balanced vanilla peppermint flavor. Peppermint flavors can vary in intensity. If you want a stronger peppermint flavor, increase the amount of peppermint flavor to your desired level by tasting it. Nutrition values are estimates.