Preheat your oven to 175°C (350°F). Grease a 9-inch round cake pan and line the bottom with parchment paper. Set aside.
In a heatproof bowl set over a pot of barely simmering water (or in the microwave in 30-second bursts), melt the dark chocolate and butter together, stirring until smooth. Remove from heat and let cool for 5 minutes.
In a large bowl, whisk together the sugar and eggs vigorously for about 1 to 2 minutes, until the mixture is slightly pale and thickened. Add the vanilla extract and whisk again.
Pour the cooled chocolate mixture into the egg mixture and stir to combine. Sift in the flour, cocoa powder, salt, and baking powder. Fold gently until just combined — do not overmix.
Pour the batter into the prepared pan and smooth the top. Bake for 30 to 35 minutes, until the edges are set and the center has a slight jiggle but is no longer liquid. A toothpick inserted in the center should come out with moist crumbs.
Let the cake cool in the pan for at least 30 minutes before removing. Slice and serve as-is, or with your toppings of choice.
Notes
For an extra fudgy result, bake closer to 30 minutes. Store covered at room temperature for up to 3 days, or refrigerate for up to 5 days. Bring to room temperature before serving for best texture.