With the mixer on low speed, add the butter, a few tablespoons at a time. Once incorporated, increase the mixer speed to medium-high and beat until the buttercream is silky smooth, 3 to 5 minutes. Use a rubber spatula to scrape down the paddle and bottom and sides of the bowl. With the mixer on medium-low, add the peppermint extract, ½ teaspoon at a time, tasting the buttercream after every addition until satisfied with the flavor.