In a bowl, whisk together the flour and salt. In a large bowl, using an electric mixer on medium-high speed, beat the butter and sugar until light and fluffy, 2-3 minutes. Add the egg whites and peppermint extract and beat on low speed until the egg is completely incorporated. Beating on low speed, slowly add the flour mixture and continue to beat until just incorporated, scraping down the sides of the bowl as needed.
Place 2 tablespoons of the dough into each of 4 small bowls. Color each a different shade of red, starting with pale pink and ending with dark burgundy. To make burgundy, use the tip of a toothpick to add a tiny bit of brown to deep red. Spoon the doughs into separate pastry bags, each with a 1⁄8-inch (3-mm) round tip. Refrigerate the dough until chilled, about 30 minutes.
Preheat the oven to 325°F (165°C). Line 3 baking sheets with parchment paper. To form each cookie, use a small offset spatula to spread 2 teaspoons of the uncolored dough on the prepared sheets into a 3.5 by 4.5-inch (9-by-11.5-cm) rectangle about 1/8 inch thick. Pipe a diagonal line of red dough over the rectangle from one side to the opposite side. Continue to pipe lines, using different colors of red and varying the width of the lines. Space the lines 3/4 inch (2 cm) apart.
Bake until the edges are firm to the touch but have not yet taken on any golden color, 15-17 minutes. Immediately use a metal spatula to remove each cookie from the sheets and invert it onto a work surface. Roll the cookie around a chopstick, then remove the chopstick. Transfer to wire racks and let cool completely, about 15 minutes. If the cookies become too cool and fragile, return them to the oven for 1 minute to soften. Store the cookies in an airtight container for up to 2 days.