Preheat oven to 350 degrees. Line an 8-inch square baking dish with parchment paper, leaving a 2-inch overhang on 2 sides. Coat with cooking spray; set aside.
Melt together butter, chocolate, and cocoa in a heatproof bowl set over a pan of simmering water. Remove from heat. Stir until smooth; let cool slightly.
Whisk together flour, baking powder, and salt in a medium bowl; set aside. Put eggs, sugar, and vanilla into the bowl of an electric mixer fitted with the whisk attachment. Beat until pale, about 4 minutes. Beat in chocolate mixture. Add flour mixture; beat until just combined, scraping down sides of bowl as needed.
Transfer batter to prepared pan, and smooth top with an offset spatula. Bake until a cake tester inserted into brownie between edge and center comes out with a few crumbs, 30 to 35 minutes. (Do not overbake.) Let cool 15 minutes; lift out of pan, and transfer to a wire rack. Let cool completely. Cut into 8 rectangles. Brownies can be stored in an airtight container at room temperature up to 3 days.