Place the butter in a small saucepan over medium heat. Cook for 4 to 6 minutes, stirring frequently, until the butter turns an amber color and brown bits appear on the bottom.
Remove from heat, cool for 5 minutes, then transfer to a small container. Refrigerate for about 1 hour until opaque and creamy.
In a large bowl, beat the chilled brown butter, brown sugar, and granulated sugar together for 3 minutes until sandy and well combined.
Add the sourdough starter, egg, and vanilla extract. Mix until fully incorporated.
Add the bread flour, all-purpose flour, salt, baking powder, and baking soda. Stir just until no dry streaks remain.
Fold in 1 cup (170g) of the dark chocolate and 1 cup (170g) of the milk chocolate. Save the remaining chocolate for later.
Scoop the dough into 18 portions, about 2 tablespoons each. Place them on a parchment-lined baking sheet. Cover tightly and refrigerate for at least 8 hours or overnight.
Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper. Place the reserved chopped chocolate in a small bowl.
Roll each chilled dough scoop into a smooth ball. Dip the tops into the reserved chocolate pieces. Arrange them on the baking sheets about 3 inches apart.
Bake for 18 to 22 minutes until the edges are golden but the centers are soft. For food safety, ensure the internal temperature of the cookies reaches 160°F (71°C) due to the egg.
Remove from the oven, sprinkle with flaky sea salt, and cool on the pans for 10 minutes before transferring to a wire rack.