Start by browning the butter. Melt the butter in a small saucepan over medium heat, and continue heating the butter, stirring frequently for about 7-10 minutes. As the butter cooks, it will start to bubble and foam. Continue cooking and stirring until the butter turns golden brown with little browned bits on the bottom of the pan. Remove from the heat and transfer to a bowl - you should have about 185-190 grams of brown butter. Make sure you get all those browned bits! Place in the freezer to cool to the touch for about 10-15 minutes.
Add the brown sugar, granulated sugar, and cooled brown butter to a large bowl. Whisk for a minute until fluffy and lighter in color. Add in the egg, egg yolk, sourdough discard, and vanilla. Whisk for another minute until well combined.
Whisk the flour, baking soda, and salt in a separate bowl. Add the flour mixture and chopped chocolate to the bowl with the wet ingredients. Using a rubber spatula, fold the dry ingredients into the wet ingredients until just combined. Do not overmix.
Cover the cookie dough and refrigerate overnight or at least 1 hour. (Please note that chilling the dough allows the flour to fully hydrate, the butter to solidify and the flavors to come together. If you bake room temperature dough, the cookies will spread too much.)
When you’re ready to bake, remove the dough from the fridge to warm up skightly, for about 15 minutes, while you preheat the oven to 350°F. Line baking sheets with parchment paper.
Scoop 3 tablespoons of dough at a time (I like using a cookie scoop) and place the dough balls on the parchment-lined baking sheet, spacing them 2-3 inches apart. If desired, you can add a few pieces of chopped chocolate to the tops of each cookie.
Bake for 10-11 minutes until the edges are lightly golden and the centers still look gooey and pale. Remove from the oven and immediately sprinkle with flakey sea salt. Let the cookies rest for 2-3 minutes before transferring to a cooling rack.
