Line baking sheets with parchment paper and preheat the oven to 350°F (175°C).
In a large bowl, whisk together the melted and slightly cooled butter, packed light brown sugar, and granulated sugar until smooth and well combined.
Add the eggs one at a time, mixing well after each addition, then whisk in the milk and vanilla until the mixture is smooth and slightly lighter in color.
In a separate bowl, whisk together the all-purpose flour, salt, baking soda, baking powder, and cornstarch until evenly blended.
Gradually add the dry ingredients to the wet ingredients, stirring just until combined; do not overmix.
Fold in the semi-sweet chocolate chips and the roughly chopped milk chocolate, reserving a few pieces to press onto the tops of the dough balls.
Chill the dough for at least 30 minutes, or ideally refrigerate 24–72 hours for best flavor and texture.
Scoop chilled dough using a 3-tablespoon cookie scoop onto the prepared baking sheets, spacing the cookies apart; press reserved chocolate pieces onto the tops.
Bake for 7–9 minutes, or until the edges are lightly golden while the centers still look slightly underbaked.
Remove from the oven and let the cookies cool on the baking sheet for 5 minutes before transferring to a wire rack to finish cooling.