Preheat oven to 350 degrees. Lightly butter a 9-inch square baking pan and line with parchment paper, leaving a 2-inch overhang on all sides. In a large microwave-safe bowl, combine butter, bean puree, and chocolates and microwave in 30-second increments, stirring each time, until melted and smooth, about 2 minutes. Whisk in sugar, eggs and egg white, and vanilla until smooth.
In a small bowl, whisk together flour, cocoa powder, baking powder, and salt. Stir into chocolate mixture until just combined. Pour batter into pan; smooth top. Bake until top is cracked and a toothpick inserted in center has moist crumbs attached, about 25 minutes (center will seem undercooked but will set further as it cools). Let cool completely in pan on a wire rack. Using parchment, lift brownie from pan and cut into 16 squares.