Heat oven to 350ºF. Line a greased 9-by-13-inch pan with foil or parchment allowing it to hang over the sides. Set aside.
Put butter, chocolate, and cocoa in a medium microwave-safe bowl and microwave until butter is melted. Stir, microwaving for 30 more seconds if needed, until chocolate is melted and mixture is smooth. Let cool slightly.
Whisk together flour, baking powder and salt and set aside.
Put sugar, eggs and vanilla in a large mixing bowl and beat on medium-high speed until pale, 2 to 3 min. Add chocolate mixture; beat until combined. Add flour mixture; beat, scraping down sides of bowl, until well incorporated.
Pour batter into prepared pan; smooth top with a spatula. Bake until a toothpick inserted into the brownies comes out with a few crumbs, but is not wet, 27 to 30 min (don't over bake). Let cool in pan, then lift out brownies. Cut into squares, or frost and then cut.
• For a darker chocolate taste, swap the semisweet chocolate for bittersweet chocolate. the brownies have enough sugar to keep them sweet with darker chocolate. • Store in an airtight container at room temperature for up to three days. • Frost using my recipe and top with chopped nuts if desired.