1. Preheat oven to 350°F. Line two sheet pans with parchment paper.
2. Place butter in a medium saucepan over medium heat. Cook, swirling and whisking, until melted, foaming, and golden brown with a nutty aroma, about 3-4 minutes.
3. Weigh out 110 g of the brown butter into a medium bowl. Let cool for 15-20 minutes.
4. Add brown sugar and granulated sugar to the cooled brown butter. Mix to combine.
5. Add egg and whisk well for 1-2 minutes until lightened and thickened.
6. Mix in vanilla.
7. Add flour, baking powder, baking soda, and salt. Stir with a spatula until just combined, with some flour streaks remaining.
8. Fold in chopped chocolate.
9. Scoop dough into 6 balls (about 55 g each) onto prepared sheet. Flatten slightly, press extra chocolate pieces on top, and roll into balls if desired. Keep remaining dough covered.
10. Bake for 11-13 minutes until edges are set. Remove from oven and bang pan briefly on the counter to help deflate cookies. Use a round cutter to shape if desired.
11. Sprinkle with flaky sea salt. Cool on pan for 10 minutes, then transfer to a wire rack.
12. Repeat with remaining dough. Store in an airtight container at room temperature for up to 3 days.