Brown the butter in a saucepan over medium heat until amber and nutty, about 5-7 minutes. Pour into a bowl, scrape up all the browned bits, and cool 10 minutes. Preheat oven to 375°F (190°C) and line two baking sheets with parchment.

Whisk brown sugar and granulated sugar into the cooled butter. Add egg, egg yolk, and vanilla; whisk until pale and glossy, about 1 minute.

Fold in flour, baking soda, and salt until a few streaks remain, then fold in the chocolate until no dry spots are visible.

Scoop 1/4 cup portions, roll into tall balls, and place on a prepared sheet. Cover and refrigerate at least 1 hour or up to 24 hours.

Place 4 chilled dough balls on the second sheet 3 inches apart. Bake 12-14 minutes, rotating halfway, until edges are golden and centers look slightly underbaked. Cool on the sheet 5 minutes, then transfer to a rack.