Cook the salted butter in a saucepan over medium heat for 8 to 10 minutes, stirring or swirling the pan as needed, until the milk solids turn golden brown. The bubbling will become quieter and the butter will smell toasted and nutty; pour it into a heatproof bowl right away and cool for 20 to 30 minutes, until it reaches 90 to 93°F.

Whisk the flour, baking soda, and salt together in a medium bowl. Blend until the leavening and salt are evenly dispersed, with no streaks of baking soda remaining.

Stir the cooled brown butter with the light brown sugar and granulated sugar in a large bowl using a spatula. The mixture should look thick and damp, similar to wet sand, rather than glossy or runny.

Mix in the egg, milk, and vanilla extract until the mixture looks creamy and unified. Stop once it is smooth so the dough does not become overly aerated.

Fold in the flour mixture until only a little dry flour remains, then fold in the dark chocolate chunks and semisweet chocolate chips. Cover the dough and refrigerate it for 1 to 4 hours, until it feels firmer and less greasy.

Heat the oven to 375°F and line two large cookie sheets with parchment paper. The hotter oven helps the outside set quickly, which supports crisp edges around soft centers.

Portion the dough into 14 pieces with a 1.35-oz cookie scoop, roll each into a smooth ball, and arrange them 3 inches apart on the prepared sheets. Shape the balls slightly taller than wide, then bake for 8 to 10 minutes, until the tops are golden and the centers still look soft.

Leave the cookies on the baking sheets for 2 minutes, then move them to a wire rack and finish with flaky sea salt if desired. The centers will settle as they cool into a dense, chewy texture.