Preheat your oven to 180°C (350°F) and line a square baking pan (about 8x8 inches or 20x20 cm) with parchment paper.
In a heatproof bowl, melt the dark chocolate and unsalted butter together. You can do this by placing the bowl over a pot of simmering water (double boiler) or by using short bursts in the microwave. Stir until smooth and set aside to cool slightly.
In a separate bowl, whisk the eggs, granulated sugar, and vanilla extract together until well combined.
Slowly pour the melted chocolate mixture into the egg mixture, whisking continuously to combine.
In another bowl, sift together the all-purpose flour, cocoa powder, baking powder, and a pinch of salt.
Gradually add the dry ingredients to the chocolate mixture, stirring until the batter is smooth and well combined.
Pour the brownie batter into the prepared baking pan and spread it out evenly.
Bake in the preheated oven for about 25-30 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs attached. Be careful not to overbake; you want the brownies to be slightly gooey in the middle.
Remove the brownies from the oven and let them cool in the pan for a while before transferring them to a wire rack to cool completely.