Preheat oven to 425°F (220°C) for at least 20 minutes. Brush ramekins with softened butter, dust with cocoa powder, tap out excess. Refrigerate on a baking sheet.
Melt chocolate and butter together in 30-second microwave intervals, stirring after each, until completely smooth. Cool for 5 minutes.
Whisk eggs, egg yolks, and sugar for 90 seconds until pale and slightly thickened. Add vanilla and whisk 10 seconds more.
Pour warm chocolate into egg mixture in a slow stream, folding gently. Sift in flour and salt; fold with 10 to 12 strokes until no streaks remain. Divide evenly among 4 ramekins, filling ¾ full.
Bake on center rack 11 to 13 minutes until edges are set and center has a slight jiggle. Run a knife around each ramekin, invert onto a warmed plate, wait 10 seconds, lift ramekin. Serve immediately.