Position oven racks in the upper and lower-middle positions and preheat the oven to 225°F (107°C). Line two large baking sheets with parchment paper.
In the clean, dry bowl of a stand mixer or a large bowl with an electric hand mixer, combine the room-temperature egg whites and cream of tartar. Beat on medium speed until the whites become frothy, about 2-3 minutes.
Increase the mixer speed to medium-high. Gradually add the sugar, about one tablespoon at a time, allowing it to dissolve before adding the next. This should take several minutes.
Once all sugar is incorporated, turn the speed to high and beat until the meringue is thick, glossy, and holds stiff peaks. When you lift the beater, the peak should stand straight up. On low speed, mix in the peppermint extract.
Using a small, clean food-safe paintbrush, paint 3 to 4 thin vertical stripes of red gel food coloring inside a large piping bag fitted with a star tip (like a Wilton 1M).
Carefully transfer the meringue into the piping bag. Pipe 1-inch kisses onto the prepared baking sheets, spacing them about 1 inch apart.
Bake for 1 hour, rotating the sheets halfway through. After 1 hour, turn the oven off completely. Leave the meringues in the closed oven to dry for at least 1 more hour, or up to 3 hours. Do not open the oven door during this time.
The cookies are ready when they feel very light, are crisp all the way through, and peel easily off the parchment paper.