Preheat your oven to 375°F (190°C). If you use a convection oven, drop the temperature to 350°F (175°C). Position two racks in the upper and lower thirds of the oven.
Brown the butter. In a light-color skillet or saucepan over medium heat, melt the butter, swirling the pan occasionally. It will foam, crackle, and then turn a deep amber. You'll smell a distinct toasted-nut aroma—that's the Maillard reaction doing its beautiful work. Don't walk away; brown butter goes from perfect to burnt in seconds. Once it's amber with dark specks at the bottom, pour it into a large mixing bowl and let it cool for 15 minutes.
Whisk the dry ingredients. In a medium bowl, whisk together the flour, baking soda, and salt.
Mix the sugars and butter. Add the brown sugar and granulated sugar to the cooled brown butter. Stir until the mixture looks like wet sand and no butter pools remain. It won't be fluffy, and that's okay.
Add the egg and vanilla. Add the egg, then the yolk, then the vanilla. Stir until glossy and fully combined; the texture will be slick and almost shiny.
Fold in the dry ingredients. Add the flour mixture and fold with a spatula or wooden spoon just until you no longer see streaks of flour. Overmixing develops gluten and makes cookies tough.
Add the chocolate. Reserve a few chocolate chunks or chips to press onto the tops of the dough balls. Fold the rest in.
Chill the dough. Cover the bowl and refrigerate for at least 2 hours—24 to 72 hours gives you a dramatically deeper flavor and a thicker cookie. Trust me on this one.
Scoop and bake. Scoop 2-tablespoon portions of dough (a #40 cookie scoop works perfectly) onto parchment-lined baking sheets, spacing them 3 inches apart. Press a few reserved chocolate pieces onto the top of each ball.
Bake, then rest. Bake for 11 to 13 minutes, rotating the sheets halfway through. The edges should look deep golden brown while the centers still appear puffy and slightly wet. Let the cookies cool on the baking sheet for 5 minutes—they'll continue to set—then transfer them to a wire rack. Sprinkle with flaky sea salt while they're still warm.