Preheat the oven to 200 degrees F. Line two baking sheets with parchment paper and set aside.
In the bowl of a stand mixer, using the whisk attachment, beat the egg whites until foamy. Add the cream of tartar and beat until soft peaks form.
On high speed, add the powdered sugar, one spoonful at a time. Add the peppermint extract. Keep beating until stiff, glossy peaks form.
Using a piping bag and open star tip, pipe swirls/roses onto the prepared baking sheets. You can also use two spoons to dollop the meringue onto the baking sheets.
Bake in the preheated oven for 3 hours, rotating the baking sheets halfway through. Turn off the heat, crack the oven door, and allow them to cool for 2 hours in the oven.
Spread melted dark chocolate on the bottom of one meringue and top it with another meringue.
Store the cookies in an airtight container at room temperature.