Brown the butter in a saucepan over medium heat until nutty and speckled. Cool for 20 minutes.
Beat the cooled brown butter with the dark brown and granulated sugars for 2 minutes.
Add the egg, egg yolk, and vanilla extract. Beat on medium-high for 3 minutes until pale and thick.
Whisk together the flour, baking soda, and kosher salt. Fold into the wet ingredients until just combined.
Cover and chill the dough in the refrigerator for at least 24 hours.
Preheat oven to 350°F (175°C). Scoop 3-tablespoon balls of dough onto lined baking sheets. Sprinkle with flaky sea salt.
Bake for 11-13 minutes until edges are golden but centers are still soft.
Cool on the baking sheet for 10 minutes before moving to a wire rack.