Preheat oven to 350 degrees. Butter 9-by-13-inch cake pan; line with parchment paper, allowing 2-inch overhang. Butter parchment. Put butter, chocolate, and cocoa in heatproof bowl set over pan of simmering water; stir until butter and chocolate have melted. Let cool slightly.
Whisk flour, baking powder, and salt in medium bowl. Put sugar, eggs, and vanilla into mixer bowl; whisk on medium until pale, about 4 minutes. Mix in chocolate mixture. Reduce speed to low. Add flour mixture; mix until just combined.
Spread batter into prepared dish; smooth top. Bake until a cake tester inserted into center comes out with only a few moist crumbs attached, 35 to 40 minutes. Let cool in dish 40 minutes. Lift out, and let cool completely on wire rack.