Preheat oven to 175 degrees. Line 2 baking sheets with parchment paper; secure corners with masking tape. Fit a pastry bag with a small open-star tip. Set aside.
Make meringues: Put egg whites and sugar in the heatproof bowl of an electric mixer. Set bowl over a pan of simmering water, and stir gently until sugar has dissolved and mixture is warm to the touch, 2 to 3 minutes.
Transfer bowl to an electric mixer fitted with the whisk attachment. Mix on medium-high speed until stiff peaks form. Mix in peppermint extract.
Using a new small paintbrush, paint 2 or 3 stripes of red food coloring inside the pastry bag. Fill bag with 1 to 2 cups meringue. Pipe small (3/4-inch-high) star shapes onto prepared baking sheets. Refill bag as necessary, adding food coloring each time.
Bake cookies until crisp but not brown, about 2 hours. Let cool completely on sheets on wire racks.