Whisk the meringue mixture occasionally over the simmering water until it reaches 160 F, or until the sugar has completely melted and the mixture is hot to the touch — just about too hot to hold your finger in. Once it's reached this stage, transfer the mixture to an electric mixer fitted with a whisk attachment and whisk on a low setting until the bottom of the bowl is cool to the touch, about 10-15 minutes.
Increase the mixer speed to high and continue to whisk the meringue until you achieve stiff peaks. Once the peaks are firm, add the butter a little at a time (about 2-3 tablespoons), allowing each addition to be fully incorporated before adding more. If the mixture appears to curdle, simply continue mixing on high speed; it will eventually come together.
Once the icing is smooth, add 2-3 teaspoons of peppermint extract and tint it mint green with food coloring. Then, pipe or spread the icing onto your cooled cupcakes.