Preheat the oven to 275 degrees . Line 2 baking sheets with parchment paper.
In a food processor, pulse the sugar until finely ground and transfer to a bowl. Pulse the peppermint candies until finely chopped; transfer to a measuring cup. Return any large pieces to the processor and pulse a few more times. (You should have a scant 1/2 cup crushed candy.)
Using an electric mixer, beat the egg whites and cream of tartar on medium-high speed until soft peaks form. Slowly add the sugar and continue beating until the whites form stiff glossy peaks. Fold in the crushed peppermints. Drop the mixture by teaspoonfuls about 1 inch apart onto the baking sheets.
Bake the cookies for 25 minutes. Reduce the oven temperature to 250 degrees and bake for 25 minutes longer. Let cool completely on a wire rack.