Melt butter in a light-colored saucepan over medium heat, swirling occasionally. Cook 3 to 4 minutes until it foams and turns golden brown with a nutty aroma and dark flecks on the bottom. Immediately pour into a large heatproof bowl, scraping in all the browned bits. Let cool 10 minutes.

Whisk brown sugar and granulated sugar into the cooled brown butter until well combined and slightly glossy. The mixture will look like wet sand.

Whisk in eggs one at a time until fully incorporated, then whisk in vanilla. The dough should be smooth and thick.

In a separate bowl, whisk together flour, baking soda, and kosher salt. Fold into the wet mixture with a rubber spatula until just combined with no dry streaks remaining.

Fold in chocolate chips. The dough will be thick and slightly greasy from the brown butter. Let rest at room temperature 10 minutes while you heat the oven.

Heat oven to 375 degrees F. Line baking sheets with parchment. Scoop 2-tablespoon mounds of dough, spacing 2 inches apart. Bake 10 to 12 minutes until edges are set and golden but centers still look slightly underdone.

Cool on the baking sheet 5 minutes, then transfer to a wire rack. Sprinkle warm cookies with flaky sea salt. Cool completely or enjoy warm.