Start by preparing your pan and preheating the oven. Grease and line the base and sides of a 23cm (9-inch) springform pan. Preheat the oven to 180°C (350°F).
Whisk the eggs and sugars. In a medium-sized bowl, combine the eggs, light brown sugar, granulated sugar, and vanilla extract. Using a hand-held electric whisk, beat on medium-high speed for about 4 minutes, until the mixture becomes light and creamy. When lifting the whisk, the mixture should leave trails for a few seconds.
Add the dry ingredients and fold in. Sift the flour, cocoa powder, baking powder, and espresso powder into the egg mixture, then add the salt. Mix on low speed until just a few streaks of flour remain.
Incorporate the melted chocolate. Add the melted dark chocolate into the batter and fold gently with a rubber spatula. Be sure to scrape the bottom of the bowl as you fold to ensure the ingredients are fully combined and smooth.
Bake the batter. Pour the batter into the prepared springform pan. Tap the pan firmly on the work surface a few times to remove any air bubbles, and use an offset spatula if necessary to smooth the surface.
Bake for 15-20 minutes, or until the center springs back when lightly touched.
Cool the sponge. Remove from the oven and set the pan on a wire rack to cool to room temperature. The top of the sponge will likely appear wrinkled. To flatten it, press the top down gently with a flat-bottomed glass or the base of a measuring cup. Once flat, cover and refrigerate until you are ready to proceed with the next step.