Preheat your oven to 425°F (220°C). Grease 4 (6-ounce) ramekins generously with butter, then dust thoroughly with cocoa powder, tapping out any excess. Place the prepared ramekins on a baking sheet.

In a double boiler or a microwave-safe bowl, combine the 1/2 cup unsalted butter and finely chopped dark chocolate. Melt gently, stirring occasionally, until smooth and fully combined. If using a microwave, heat in 30-second intervals, stirring after each, to prevent burning. Set aside to cool slightly.

In a medium mixing bowl, whisk together the large eggs, egg yolks, granulated sugar, and vanilla extract until light in color and slightly thickened, about 2-3 minutes. This aerates the mixture and helps with the cake's texture.

Gradually pour the slightly cooled chocolate mixture into the egg mixture, whisking constantly until just combined. Do not overmix. Gently fold in the all-purpose flour and salt with a rubber spatula until no streaks of flour remain.

Divide the batter evenly among the prepared ramekins. Place the baking sheet with ramekins into the preheated oven. Bake for 11-13 minutes, or until the edges are set and slightly firm, but the center still jiggles when gently shaken. The cakes should look slightly puffed.

Carefully remove the baking sheet from the oven. Let the cakes rest in the ramekins for 1 minute. Place a serving plate upside down over each ramekin, then invert the ramekin and plate together. Gently lift the ramekin to release the cake. Dust with confectioners' sugar and serve immediately with fresh berries or a scoop of vanilla bean ice cream, if desired.