Preheat & Prep Ramekins: Preheat your oven to 425°F (220°C). Generously grease four 6-ounce ceramic ramekins with melted butter, then dust the interiors evenly with cocoa powder, tapping out any excess. Place the ramekins onto a sturdy rimmed baking sheet.
Melt Chocolate & Butter: In a heatproof bowl set over a pot of barely simmering water (double boiler) or in the microwave in 30-second bursts, melt the chopped dark chocolate and cubed butter together. Whisk gently until glossy, smooth, and completely emulsified. Remove from heat and stir in vanilla extract and espresso powder; set aside to cool slightly.
Whisk Eggs & Sugar: In a medium mixing bowl, whisk together the 2 whole eggs, 2 egg yolks, powdered sugar, and fine salt vigorously with a wire whisk for about 1 minute until pale, slightly thickened, and smooth.
Combine & Fold Flour: Pour the melted chocolate mixture into the egg mixture and whisk gently until fully incorporated. Sift the all-purpose flour over the batter and fold gently with a silicone spatula just until no dry flour streaks remain. Do not overmix.
Divide & Bake: Divide the glossy chocolate batter evenly among the four prepared ramekins (about 3/4 full). Bake at 425°F for exactly 11 to 13 minutes. The edges should look firm and slightly puffed, but the center will still have a gentle, soft jiggle when nudged.
Invert & Serve: Remove from the oven and let rest on the baking sheet for exactly 1 minute. Run a small butter knife gently around the top edges. Invert a dessert plate over each ramekin, carefully flip together (using an oven mitt), and gently lift the ramekin away. Dust immediately with powdered sugar, sprinkle with flaky sea salt, and top with a scoop of vanilla ice cream. Serve warm!