In a medium bowl, whisk together the flour, cornstarch, baking soda, and salt until evenly combined; set aside.
In a large mixing bowl, beat the softened butter with the brown sugar and granulated sugar on medium speed just until creamy and slightly lightened, about 1–2 minutes.
Beat in the vanilla and peppermint extract. Add the egg and the yolk one at a time, mixing for 30 seconds after each addition and scraping down the bowl once.
Add the dry ingredients in two additions on low speed, stopping when a few flour streaks remain. Fold in the semi-sweet chips, white chips, and peppermint crunch chips by hand with a spatula until just distributed.
Cover the bowl and chill the dough for 20 minutes. While it chills, position a rack in the center of the oven, preheat to 365°F (185°C), and line two baking sheets with parchment paper.
Scoop the dough into 2-tablespoon portions (about 1.5 oz/43 g each), roll into smooth balls, and arrange 2 1/2 inches apart on the prepared sheets.
Bake one sheet at a time for 10–12 minutes, rotating the pan halfway through. The cookies are done when edges are set and the centers look pale, puffy, and just dry on top.
Right out of the oven, tap the pan gently once on the counter to settle. Press a few extra chocolate chips into the warm tops and immediately sprinkle with crushed peppermint. Cool on the sheet for 5 minutes, then transfer to a rack to cool completely.