Combine egg whites and sugar in a large mixing bowl.
Put the mixing bowl over a double boiler. Whisk the mixture until the temperature reaches 150°F. By that point, the sugar has dissolved completely.
Transfer the egg mixer into a stand mixer with a whip attachment. Beat on high for about 5 minutes, until stiff peaks form. The bowl shouldn't be hot to the touch.
Add the butter in cubes and continue to beat on high. Add the butter one piece at a time, but you don't need to let it fully incorporate before adding the next one. It will look like it curdles at some point, but it will come back together!
Beat for about 5 minutes, until the texture is light and fluffy. Switch to the paddle attachment and add salt, vanilla and peppermint extracts. Beat for another 30 seconds to combine.