Place the butter in a light-colored saucepan over medium heat. Cook, swirling the pan often, until the butter melts, foams, and then brown bits form at the bottom, releasing a nutty aroma. This takes about 5 to 8 minutes. Immediately pour the browned butter into a heatproof bowl to stop cooking. Let it cool for 15 minutes.
In a large bowl, whisk the cooled brown butter with the light brown sugar and granulated sugar until fully combined.
Beat in the egg, egg yolk, and vanilla extract until the mixture is smooth and creamy.
In a separate bowl, whisk together the flour, baking soda, and kosher salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Fold in the chocolate chips.
Cover the dough and chill in the refrigerator for at least 30 minutes, or up to 24 hours for deeper flavor.
Preheat your oven to 375°F (190°C). Line baking sheets with parchment paper.
Scoop rounded tablespoons of dough onto the prepared sheets, leaving space between them.
Bake for 10 to 12 minutes, or until the edges are golden brown and the centers look slightly underbaked.
Remove from the oven. Immediately sprinkle the tops of the cookies with flaky sea salt.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
Notes
Browning the butter correctly is key; watch for the milk solids to turn deep brown and smell nutty.
Chilling the dough develops the flavor and prevents the cookies from spreading too thin during baking.
Use a mix of chocolate chips and chopped chocolate chunks for varied texture.