In a medium bowl, whisk together flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat butter and sugar until fluffy. Add eggs one at a time, scraping between each addition. Add peppermint extract and Greek yogurt, and mix well. Gradually add flour mixture, beating just until combined.
Divide the mixture in half, and flatten into a disk. Wrap each disk in plastic wrap. Refrigerate at least 2 hours, or overnight.
Preheat oven to 425°F. Line baking sheets with parchment paper or silicone liners.
Lightly flour a work surface. Roll dough to 1/4-inch thickness. The dough will be a little sticky, so use flour as needed to avoid sticking. Use a 2 to 2-1/2 inch round cookie cutter to cut out circles and transfer to the prepared baking sheets. Bake cookies until just set and still pale, about 7 minutes. Transfer to a cooling rack or a piece of waxed paper to cool completely before frosting.