Move rack to the bottom of oven and preheat oven to 175 degrees F (80 degrees C). Line baking sheets with aluminum foil.
Whisk egg whites and 1/4 cup white sugar in the top of a double boiler set over simmering water until smooth. Whisk cream of tartar and peppermint extract into egg mixture and beat until foamy. Gradually whisk confectioners' sugar into mixture, beating until egg white meringue holds stiff peaks.
Scoop about 1 tablespoon of meringue per cookie and gently transfer to the prepared baking sheets; use spoon to shape the meringue cookies into football shape with 2 gently pointed ends.
Bake cookies on bottom rack of the preheated oven until dry on the outside but still slightly soft in the centers, about 1 1/2 hours. Let cool completely before removing from baking sheets.
Melt chocolate chips in the top of a double boiler over simmering water until melted, stirring until smooth. Gradually mix cream into chocolate until mixture is thick but not pasty; stir 1 teaspoon white sugar into chocolate ganache until sugar has dissolved. Dip cookies in ganache. Place dipped cookies atop a piece of parchment paper and sprinkle with crushed peppermint candy. Let cookies set; refrigerate leftovers.