Preheat oven to 350°. Butter an 8″square baking pan. Line pan with parchment paper, leaving 2 inches hanging over the sides. Butter parchment paper.
In a heatproof bowl over a pan of simmering water, stir together butter, chocolate, and cocoa powder. Set aside to cool slightly.
Whisk together flour, baking powder, and salt. Set aside.
Combine sugar, eggs, and vanilla using an electric mixer with a whisk attachment. Mix on medium speed until pale, about 4 minutes. Add chocolate mixture and mix until combined. Reduce speed to low and add flour mixture, mixing until combined.
Pour batter into prepared pan. Spread evenly with spatula. Bake about 35 minutes, or until a cake tester comes out with a few crumbs but is not wet. Let cool in pan about 15 minutes.
Lift brownies out of pan. Cool completely on a wire rack before cutting into squares.
Makes 9 large or 16 small brownies