Step 1: Brown the butter. In a light-colored saucepan, melt the unsalted butter over medium heat. Continue to cook, swirling occasionally, until the butter foams, then subsides, and golden-brown bits appear at the bottom. It will smell nutty. Immediately pour the brown butter into a heatproof bowl to stop the cooking and let it cool slightly for about 10-15 minutes.
Step 2: In a large mixing bowl, combine the slightly cooled brown butter, granulated sugar, and light brown sugar. Whisk well until smooth and fully incorporated.
Step 3: Add the egg and then the egg yolk, one at a time, mixing thoroughly after each addition until combined. Stir in the vanilla extract.
Step 4: In a separate medium bowl, whisk together the bread flour, baking soda, and salt, ensuring there are no lumps.
Step 5: Gradually add the dry ingredients to the wet ingredients, mixing on low speed with a stand mixer or by hand with a spatula until just combined. Be careful not to overmix the dough.
Step 6: Fold in the dark chocolate chunks and milk chocolate chunks until evenly distributed throughout the dough.
Step 7: Cover the dough tightly with plastic wrap and refrigerate for a minimum of 2 hours, or preferably overnight (up to 24-48 hours) for best flavor and texture development.
Step 8: When ready to bake, preheat your oven to 375°F (190°C). Line two baking sheets with parchment paper.
Step 9: Scoop generous portions of chilled dough, about 2-3 tablespoons each, onto the prepared baking sheets, leaving at least 2 inches between cookies. For a more rustic look, you can tear off chunks of dough rather than rolling perfect balls.
Step 10: Bake for 10-12 minutes, or until the edges are golden brown and the centers are still slightly soft and puffy. Do not overbake. For extra visual appeal, you can gently press a few additional chocolate chunks onto the tops of the warm cookies immediately after removing them from the oven.
Step 11: Let the cookies cool on the baking sheets for 5-10 minutes before transferring them to a wire rack to cool completely. This allows them to set and develop their signature chewy texture.