In the bowl of a stand mixer fitted with the paddle attachment, combine the cold cubed butter, light brown sugar, and granulated sugar. Mix on low speed just until the butter is broken into small pea-sized pieces and coated with the sugar. Do **not** cream the butter and sugar.
Add the egg, egg yolk, and vanilla extract. Mix on low just until combined.
In a separate bowl, whisk together the flour, baking soda, baking powder, cornstarch, and salt.
Gradually add the dry ingredients to the butter mixture, mixing on low just until no dry streaks remain. Do not overmix.
Fold in the semi-sweet chocolate chips and chopped milk chocolate until evenly distributed.
If using gluten-free flour, let the dough rest for 15 minutes before chilling.
Cover and refrigerate the dough for at least **2 hours**, or preferably overnight for the best flavor and texture.
Preheat the oven to 375 degrees. Line two baking sheets with parchment paper.
Using a large cookie scoop (about ¼ cup), scoop the dough onto the prepared baking sheets. Gently shape each scoop into a tall, rough mound instead of a smooth ball. Leave about 2 inches between each cookie. For bakery-style cookies, press a few extra pieces of chopped chocolate onto the tops before baking.
Bake for 12-14 mins, or until the edges are lightly golden and the centers still look slightly underbaked.