Preheat oven to 350 F (180C). Line two baking trays with parchment paper. Have everything ready to go, because it's best to bake the batter right after mixing it.
Melt the butter and chocolate in a heatproof bowl set over a saucepan of gently simmering water. Stir occasionally until fully melted. Set aside.
In the bowl of a stand mixer fitted with the whisk attachment, whisk together the eggs and both sugars, on medium-high speed, for 5 minutes- a full 5 minutes, the eggs will thicken and become voluminous.
Meanwhile, sift together the flour, cocoa powder, baking powder and salt and set aside.
Once the egg mixture is light and fluffy and 5 minutes have passed, add the chocolate mixture and mix for a minute or so to combine.
Add dry ingredients and mix very briefly until just combined. Use a spatula to give one last mix, scraping up the bottom.
Fill a tablespoon with a heaping amount of batter, and use your finger to slide onto baking sheets. Space dough balls 2 inches apart, as they will spread. Dough will be wet, much like brownie batter.
Sprinkle with sea salt and bake 12 minutes. Cookies will be set but soft, cool completely before enjoying. I recommend baking one sheet at a time in the center of the oven, but having the dough scooped out on the baking sheets and ready. You can bake two sheets at a time, but swap their positions halfway through bake time.