One day ahead, infuse the butter: Chop the softened butter into chunks. In a food processor, pulse the fresh mint sprigs with the butter until the mint is finely pulverized and fully incorporated. Scrape into a bowl, cover, and refrigerate for 12-24 hours to infuse. Before using, let it soften at room temperature. Optionally strain through a fine-mesh sieve.

Cream the butter and sugar: In the bowl of a stand mixer fitted with the paddle attachment, combine the mint-infused butter, 100g of confectioners' sugar, and salt. Cream on medium speed for 3-4 minutes until very pale and fluffy.

Add extracts: Beat in the vanilla extract and peppermint extract until just combined.

Incorporate dry ingredients: Add the flour (and the optional minced fresh mint, if using) all at once. Mix on low speed just until the dough comes together and no dry flour remains.

Chill the dough: Divide the dough in half, shape each half into a disc, wrap tightly in plastic wrap, and refrigerate for at least 1 hour.

Preheat oven and prepare pans: Preheat your oven to 325°F (163°C). Line baking sheets with parchment paper.

Roll and cut the cookies: On a surface lightly dusted with confectioners' sugar, roll one disc of dough to a 1/4-inch thickness. Cut into rounds using a 2-inch cookie cutter, placing them 1 inch apart on the prepared baking sheets. Re-roll scraps once. Chill the cut cookies on the sheets for 20 minutes.

Bake: Bake for 12-15 minutes, rotating the sheet halfway through, until the cookies are set and the bottoms are just barely golden. The tops should not brown.

Cool: Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely.

Finish: Once completely cool, dust the cookies generously with additional sifted confectioners' sugar.