Preheat your oven to 220°C (430°F) using the conventional setting (refer to note 2 if using a fan-assisted oven), and generously grease and flour four 3.5-inch (6-ounce) ramekins. If your ramekins are slightly larger or smaller, that’s perfectly fine—just be prepared to adjust the baking time accordingly.
Place the butter and chocolate in a heatproof bowl. Melt the mixture using a double boiler or in the microwave (the microwave is often quicker and more convenient). If using a microwave, heat in 20-second intervals, stirring between each burst, until the chocolate is nearly melted. At that point, continue stirring to allow the residual heat to finish melting it completely. Once the mixture is smooth, set it aside.
In a large mixing bowl, combine the brown sugar, vanilla extract, salt, whole eggs, and egg yolks. Using either a hand mixer or a stand mixer fitted with the whisk attachment, beat the mixture on medium speed for about 5 minutes until it becomes thick and pale. The batter should be able to form ribbons that briefly sit on top before sinking back in.
Sift the all-purpose flour directly into the egg mixture, then pour in the melted chocolate and butter mixture. Using a spatula, gently fold everything together until the batter is smooth and evenly mixed.
Divide the batter evenly between the prepared ramekins, filling each about three-quarters full. Tap each ramekin gently on the countertop to release any trapped air bubbles, then place them all on a baking tray. Bake for 11 to 12 minutes, or until the edges appear set but the centers still jiggle slightly when touched. (If you're concerned about overbaking, see note 3 for guidance.)
Allow the cakes to cool in their ramekins for about one minute. Then, using a tea towel or oven mitts to protect your hands from the heat, carefully invert each ramekin onto a serving plate or saucer. Let them rest for another 15 seconds before gently lifting the ramekin—if properly greased, it should release easily.
Dust with powdered sugar just before serving. For an extra treat, add a scoop of vanilla ice cream, a dollop of whipped cream, or a handful of fresh berries on the side.