Preheat oven to 350°F. Line baking sheets with parchment paper. Set aside.

In a medium skillet or saucepan, melt the butter over medium heat. Once the butter has melted, continue cooking over medium heat, stirring and/or swirling the pan frequently. The butter will bubble, then foam up, then turn a golden brown color with a nutty aroma. (This will be easier to see if you use a light-colored pan.)

As soon as the butter is browned and nutty, remove it from the heat and pour it into a bowl, making sure to get the browned bits from the bottom of the pan. Set aside to cool for 20-30 minutes.

In a medium bowl, whisk together the flour, baking soda, and fine sea salt. Set aside.

In a large bowl, whisk together the sugars and the cooled brown butter. Add the eggs and vanilla, whisking vigorously until smooth and well combined.

Switching to a spatula or wooden spoon, stir the dry ingredients into the butter mixture until just combined. Use a large (3 tablespoon) cookie scoop to portion the dough onto the prepared baking sheets, leaving at least 3 inches between each cookie. If desired, sprinkle each cookie with a bit of flaky sea salt.

Bake for 12-14 minutes, until the cookies are golden. Let the cookies rest on the baking sheet for 3-5 minutes before moving to a wire rack to finish cooling.