In a medium saucepan, melt the butter over medium heat, stirring frequently. Continue cooking until the butter turns a deep golden brown and releases a nutty aroma, about 5–7 minutes. Watch closely to avoid burning. Once browned, transfer the butter to a heatproof bowl and let it cool for about 10 minutes.

Adjust an oven rack to the middle position. Preheat your oven to 350°F. Line two baking sheets with parchment paper to prevent the cookies from spreading too much during baking.

In a medium mixing bowl, whisk together the flour, baking soda, salt, and cornstarch. Make sure the dry ingredients are evenly combined, then set the bowl aside.

In the bowl of a stand mixer fitted with the paddle attachment, combine the cooled brown butter, granulated sugar, and brown sugar. Beat on medium speed for 2–3 minutes, until the mixture becomes light and fluffy, scraping down the sides of the bowl with a rubber spatula as needed.

Scrape down the sides and bottom of the bowl again with a rubber spatula. Add the whole egg and mix on medium speed for 15–20 seconds. Then, add the egg yolk and repeat. Scrape the sides and bottom of the bowl once more, then add the vanilla extract and milk. Mix on low speed until fully incorporated. The batter should look smooth and glossy.

Gradually add the dry ingredient mixture to the wet ingredients. Mix on low speed until a soft dough forms, stopping to scrape down the bowl as necessary. Avoid overmixing, as this can make the cookies dense.

Fold the chopped bittersweet chocolate into the dough using a rubber spatula, ensuring the chocolate is evenly distributed.

Using a 3-tablespoon-sized cookie scoop, portion all the dough into balls. Once all the dough is scooped, take two balls and press them together to create a large "bakery-style cookie," approximately a heaping ⅓ cup each. Roll the combined dough into smooth, even balls. Place four dough balls on each prepared baking sheet, spaced well apart. Transfer one baking sheet to the freezer for 15 minutes (or the fridge for 25 minutes) before baking. Keep the other sheets in the fridge while you bake the first batch.

Place the chilled baking sheet in the preheated oven and bake for 10 minutes, until the cookies are puffed and slightly set in the center. Remove the cookie sheet from the oven and bang it on the counter three times. (You don’t need to bang it too hard, but make sure it’s firm enough to create ripples in the cookies.) Return the cookies to the oven and bake for an additional 3–4 minutes, depending on your preference. Bake until the edges are golden brown and the centers are pale and slightly underbaked. For a gooey center, bake for 3 minutes; for a crisper texture, bake for 4 minutes.

Remove the baking sheet from the oven and let the cookies cool completely on the pan. The residual heat will help set the centers slightly. Once cooled, transfer the cookies to a wire rack or serve immediately for the best gooey texture.