Preheat oven to 350°F (175°C). Butter an 8-inch square baking pan. Line the pan with parchment paper, leaving 2 inches hanging over the sides. Butter the parchment paper.
In a heatproof bowl over a pan of simmering water, stir together the butter, chocolate, and cocoa powder until melted and smooth. Set aside to cool slightly.
Whisk together the flour, baking powder, and salt. Set aside.
In a separate bowl, combine the sugar, eggs, and vanilla using an electric mixer with a whisk attachment. Mix on medium speed until pale, about 4 minutes. Add the chocolate mixture and mix until combined.
Reduce mixer speed to low and gradually add the flour mixture, mixing until combined.
Pour the batter into the prepared pan and spread evenly. Bake for 35 minutes, or until a cake tester comes out with a few moist crumbs.
Let the brownies cool in the pan for about 15 minutes. Then, use the overhanging lining to lift the brownies out of the pan onto a wire rack. Cool completely before cutting into squares.
