First, brown the butter. Melt the butter in a small saucepan over medium heat. Continue cooking, swirling the pan often, until the milk solids turn golden brown and it smells nutty, about 5-7 minutes. Pour into a bowl and chill in the refrigerator until solid but scoopable, about 1 hour.
In a large bowl or the bowl of a stand mixer, beat the chilled brown butter with the brown sugar and granulated sugar until the mixture looks like wet sand, about 3 minutes.
Add the sourdough starter, egg, and vanilla extract. Mix on low speed until just combined.
In a separate bowl, whisk together the all-purpose flour, bread flour, salt, baking powder, and baking soda. Add the dry ingredients to the wet ingredients and mix on low until just a few streaks of flour remain.
Reserve 1/4 cup each of the chopped dark and milk chocolate for topping. Add the remaining chocolate to the dough and fold it in with a spatula until evenly distributed. Cover the bowl and refrigerate for at least 8 hours, or overnight.
When ready to bake, preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper.
Scoop the cold dough into balls about 2 tablespoons each (around 55g). Roll each ball between your hands to smooth it, then press the top into the reserved chopped chocolate.
Arrange the dough balls on the prepared baking sheets, leaving about 3 inches between them. Bake for 18 to 22 minutes, until the edges are golden but the centers are still soft.
Remove from the oven and immediately sprinkle with flaky sea salt, if using. Let the cookies cool on the baking sheet for 10 minutes before moving them to a wire rack to cool completely.