In a large bowl, beat together the softened butter, dark brown sugar, and granulated sugar until well combined, about 1 minute with an electric mixer. Add in the eggs, milk, and vanilla extract and mix until well combined- the mixture may look slightly split at this point but that's ok!
Add in the flour, cornstarch, salt, and baking soda, and beat together until a soft dough forms and there are no dry pockets of flour remaining.
Using a spatula, stir in the chopped chocolate, being sure to reserve about 2-3 tbs. of each type of chocolate. Scoop the dough into even cookie dough balls about 2-tablespoonfuls onto a lined baking sheet that will fit into your fridge.
Gently press a couple of chunks of the reserved chocolate into the top of each cookie dough ball, and sprinkle all the cookie dough generously with flakey salt. Cover the pan tightly with plastic wrap and chill for at least an hour, and up to 12 hours before baking.
When you're ready to bake, line a baking sheet with parchment paper and preheat your oven to 350 degrees. Place your chilled dough straight from the fridge onto your baking sheet with space for the cookies to spread- about 6 cookies per half-sheet pan. Bake the cookies for 8-10 minutes, or until the edges are lightly golden brown. The centers may look a little under baked, but they will cook through while they cool on the baking sheet. Cool the cookies completely on the baking sheet and enjoy the cookies warm. Enjoy!