Brown the Butter. Melt the butter in a light-colored saucepan over medium heat. A light pan matters here, because you need to see the color change. Swirl the pan now and then as the butter melts, foams, and then quiets. When the foam subsides, look for golden-brown flecks at the bottom and a nutty smell. That's brown butter. Pour it immediately into a heatproof bowl, scraping in every browned bit, and let it cool for 15 minutes. Hot butter will melt the sugars into a puddle and give you a greasy, thin cookie, so don't rush this.

Mix the Dough. Whisk the brown sugar and granulated sugar into the warm brown butter until the mixture looks glossy and smooth, about 1 minute. Whisk in the egg, then the yolk, then the vanilla, until the batter ribbons off the whisk. Add the flour, baking soda, and fine salt. Fold with a spatula just until the last streaks of flour disappear, then fold in the chopped chocolate and the chips. Stop there. Ten extra folds is the difference between chewy and tough, so put the spatula down.

Chill, Scoop, Chill Again (Briefly). Cover the bowl and chill at least 1 hour, up to 72 hours. Scoop the cold dough into balls of about 3 tablespoons each (a #24 scoop, roughly 55 g). If the dough got rock hard in the fridge overnight, let the tray of scooped balls sit on the counter for 15 minutes before baking. You want the balls cold, not frozen solid, going into the oven.

Bake and Underbake (On Purpose). Heat the oven to 350°F with a rack in the center. Line baking sheets with parchment. Space 8 balls per sheet, no more. Crowding the pan lets the cookies merge at the edges, and then it isn't really a batch of cookies, it's a sheet of cookie. Bake 11 to 13 minutes. The edges should look set and golden while the centers still look pale, puffy, and underdone. Pull them anyway. Carryover heat finishes the job while they sit on the pan, and that's what keeps the centers soft. If they look done in the oven, they're already overbaked. Optional but encouraged: lift the pan and drop it flat on the counter once, right out of the oven. The centers collapse into those ripples and wrinkles. Sprinkle with flaky salt while the cookies are still warm. Cool on the pan for 10 minutes, then move the cookies to a rack.