Heat the oven. Set a rack in the middle of the oven and heat it to 425°F (220°C). Give it a full 15-20 minutes so the oven is truly at temperature, not just beeping.
Prep the ramekins. Brush four 6-ounce ramekins generously with the softened butter, covering the bottom and every bit of the sides. Add the cocoa powder, tilt and tap to coat, then knock out the excess. Set the ramekins on a small rimmed baking sheet.
Melt the chocolate and butter. Put the chopped chocolate and butter in a large microwave-safe bowl and microwave on high for 1 minute, until the butter is melted. Whisk until the chocolate is completely smooth and glossy (microwave 10-15 seconds more if a few lumps remain).
Add the sugar. Whisk in the powdered sugar until no dry streaks remain. The mixture will look thick and a little grainy.
Add the eggs. Whisk in the eggs, egg yolks, vanilla and salt until the batter turns smooth and shiny, about 30 seconds.
Fold in the flour. Stir in the flour with a spatula just until you can't see any white. Don't beat it: the batter should be thick, glossy and fall from the spatula in a slow ribbon.
Fill. Divide the batter evenly among the ramekins, filling each about three-quarters full.
Bake 13-14 minutes. Start checking at 13 minutes. The cakes are ready when the sides look firm and pull slightly away from the ramekin, the top has risen and looks set around the edge, and a soft, slightly darker circle about the size of a quarter still wobbles in the center when you nudge the pan.
Rest 1 minute, then unmold. Let the cakes stand for exactly 1 minute. Run a thin knife around the edge of each ramekin, place a dessert plate upside down on top, and flip with oven mitts. Lift the ramekin straight up.
Serve right away. Dust with powdered sugar and serve within a few minutes; the center keeps cooking from residual heat.