Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper.

Place the chopped dark chocolate and butter in a heat-proof bowl set over a saucepan of barely simmering water (double boiler). Melt gently, stirring occasionally until smooth and glossy. Remove from heat and let cool slightly for 5-10 minutes.

In a large mixing bowl, using a hand mixer or stand mixer with whisk attachment, beat the eggs, granulated sugar, and brown sugar on high speed for exactly 5-7 minutes. The mixture should be thick, pale, and leave ribbons when the beater is lifted. Beat in the vanilla extract.

With the mixer on low speed, slowly pour in the slightly cooled melted chocolate mixture. Mix only until just combined; do not overmix.

In a small bowl, whisk together flour, sifted cocoa powder, baking powder, and fine salt. Sift this over the wet batter. Gently fold with a rubber spatula until just combined and no flour streaks remain. The batter will be thick like brownie batter.

Immediately scoop the batter onto prepared baking sheets using a medium cookie scoop (about 1.5 tbsp size), spacing them 2-3 inches apart. Do not flatten.

Bake one sheet at a time for 10-12 minutes. They are done when puffed, the tops are shiny and cracked, but the centers still look very soft and underbaked. Do not overbake.

Remove from oven and immediately sprinkle with flaky sea salt. Allow to cool completely on the baking sheets to set.