Special equipment: A pastry bag fitted with a round tip A small paintbrush or cotton swab
Position oven racks in the top and bottom thirds of the oven and preheat to 250 degrees F. Line 2 baking sheets with parchment paper.
Beat the egg whites and salt with an electric mixer on medium-high speed in a large bowl until foamy. Add the cream of tartar and continue to beat until soft peaks form. While beating, add the sugar 1 tablespoon at a time. Continue to beat until stiff peaks form. Beat in the peppermint.
Fold the top of a pastry bag fitted with a round tip over and down a few inches. Dip a paintbrush in the food coloring. Starting inside the pastry tip, brush a stroke of food coloring from the tip to the top of the bag. Repeat with 3 more evenly spaced strokes. Fill the bag with the meringue; try to spoon it directly into the center so as not to smudge the food coloring (though some smudging is inevitable). Pipe 1-inch-diameter mounds of meringue spaced 1 inch apart onto the prepared baking sheets.
Bake until the meringues are no longer glossy and feel light and dry when picked up, about 1 hour. Open the oven for a few minutes, then turn it off and shut the door. Leave the meringues in the oven until they are completely dry inside, about 2 hours. The meringues can be stored in an airtight container at room temperature for up to 2 days.
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