In a light-colored saucepan over medium heat, melt the butter and cook, swirling often, until it foams and the milk solids turn golden amber and smell nutty, 5 to 7 minutes. Pour into a large heatproof bowl, scraping in all the browned bits, and let cool until warm to the touch.

Beat the cooled butter with the brown sugar and granulated sugar until the mixture looks like wet sand, about 1 minute.

Add the egg and vanilla, and beat until glossy and smooth, about 30 seconds. Scrape the bowl.

Add the flour, baking soda, and salt. Stir gently just until no dry streaks remain, then fold in the chocolate chips.

Scoop the dough into 12 even mounds (about 2 tablespoons each) onto a plate, cover, and chill for at least 30 minutes. Preheat the oven to 350°F (175°C) and line two baking sheets with parchment.

Place the chilled dough mounds 2 inches apart on the sheets. Bake one sheet at a time for 10 to 12 minutes, until the edges are golden and the centers look slightly under. Remove from the oven, immediately sprinkle with flaky sea salt, and let rest on the hot sheet for 5 minutes before transferring to a rack.