Preheat oven to 350ºF. Line an 8-inch square baking pan with parchment paper or foil, leaving a 1-inch overhang.
In a medium bowl, sift flour, cocoa, baking powder and salt. In a medium saucepan, place butter and chocolate over lowest heat setting on stove and cook, stirring often, until melted and smooth. Set aside to cool slightly.
Whisk eggs, sugar and vanilla in a large bowl. Stir in cooled chocolate mixture. Add flour mixture and stir until just combined (do not overmix). Pour into prepared pan and bake until slightly puffed and set in center, 20 to 25 minutes. Let cool completely in pan on wire rack. Use parchment or foil overhang to lift brownies out of pan and onto cutting board to slice. Cut into squares and serve.