Place the butter in a light-colored saucepan over medium heat. Cook, stirring frequently, until the butter foams and the milk solids turn deep golden brown and smell nutty, about 5 to 7 minutes.
Transfer the browned butter, including all of the browned bits, to a large heatproof mixing bowl. Let cool for 10 to 15 minutes.
Whisk the white miso paste into the warm browned butter until completely smooth.
Add the brown sugar and granulated sugar. Whisk until glossy and fully combined.
Whisk in the egg, egg yolk, and vanilla extract until smooth.
Add the flour, baking soda, and salt. Stir just until no dry streaks remain.
Fold in the semisweet and bittersweet chocolate chunks.
Cover the bowl and refrigerate the dough for at least 2 hours, preferably overnight.
Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
Scoop the dough into 3-tablespoon portions (about 60 grams each) and place them 2 inches apart on the prepared baking sheets.
Bake for 11 to 13 minutes, until the edges are golden brown and the centers still look slightly soft.
Immediately sprinkle the tops with flaky sea salt.
Let the cookies cool on the baking sheets for 10 minutes before transferring them to a wire rack to cool completely.