Preheat the oven to 200°. Line two baking sheets with parchment paper and set aside.
In a large bowl, beat the egg whites until foamy and light. Sprinkle in the cream of tartar and beat until the egg whites are at soft peaks.
Gradually add the powdered sugar and beat until the meringue is stiff and glossy. Add the peppermint extract and mix to combine.
Put the peppermint sticks into a zip top sandwich bag and seal. Use the flat side of a meat mallet or rolling pin to smash the peppermint into very small pieces and peppermint dust.
Fold about 2/3 of the candies into the meringue to combine.
Fit a pastry bag with a large star or plain tip (note that the tip should be large enough so that candy pieces can go through the tip, unimpeded). Fill the bag with the peppermint stick meringue and pipe into pretty mounds on the parchment paper.
Sprinkle the tops of the meringue cookies with remaining crushed peppermint. Bake for 2 hours or until the meringues are set. Remove from oven and cool to room temperature. Store in an airtight container for 2-3 days.