Add the butter to a medium saucepan and melt over medium-low heat. Once melted, increase the heat slightly and continue cooking, stirring frequently. The butter will foam and sizzle as the water cooks off. Watch it closely as the bubbling begins to quiet. When the milk solids at the bottom turn golden brown and the butter smells toasted and nutty, immediately remove it from the heat. Pour the browned butter into a large mixing bowl, making sure to scrape in all the flavorful browned bits. Allow it to cool completely until it is no longer warm to the touch.

Add the brown sugar and cane sugar to the cooled browned butter and stir until well combined. Mix in the egg yolks and vanilla, followed by the sourdough discard. Stir until the mixture is smooth and evenly combined.

In a separate bowl, whisk together the flour, baking powder, baking soda, and sea salt. Gradually add the dry ingredients to the wet ingredients, stirring just until a thick cookie dough comes together. Avoid overmixing.

Roughly chop both chocolates, leaving a mixture of larger chunks and smaller pieces. Fold the dark and milk chocolate into the dough until evenly distributed.

Cover the dough and refrigerate for at least 30–60 minutes, or refrigerate overnight for a longer fermentation and more developed flavor.

Preheat the oven to 350°F (175°C) and line a baking sheet with unbleached parchment paper. Scoop the chilled dough into approximately 1½-tablespoon portions and gently roll into balls. Arrange the cookies about 2 inches apart on the prepared baking sheet. Bake for 10–12 minutes, until the edges are lightly golden but the centers still look soft and slightly underbaked.

Immediately after removing the cookies from the oven, sprinkle each one with a little flaky sea salt. Allow the cookies to cool on the baking sheet. The centers will continue to set as they cool, leaving you with crisp edges and a soft, chewy center.