Heat oven to 175°F. Spray two baking sheets with cooking spray; line with parchment paper and set aside.

Combine egg whites and sugar in the bowl of a stand mixer. Place bowl over a pan of simmering water. Stir until sugar dissolves and mixture is warm to the touch, about 3 to 5 minutes.

Transfer bowl to stand mixer; add cream of tartar and salt. Beat with whisk attachment until stiff peaks form, about 10 to 15 minutes. Add peppermint extract; briefly beat to incorporate.

Use a small paintbrush to paint three lines of red gel food coloring up the inside of a pastry bag. Fill bag with meringue mixture. Pipe 2-inch circles about two inches apart on baking sheets (12 circles per baking sheet). Pipe up the sides of the circles to make a cup about 1-inch high.

Bake meringues until they are set and can be pulled easily from the parchment, about 1 1/2 hours. Transfer to a cooling rack to cool completely.

Pipe frosting into center of meringue cups; top with crushed peppermint candies.