Step 1: Preheat your oven to 350°F (175°C). Grease and flour an 8x8 inch baking pan, or line with parchment paper, leaving an overhang on two sides to easily lift the brownies out later.
Step 2: In a large, microwave-safe bowl, combine the unsalted butter and unsweetened chocolate. Heat in 30-second intervals, stirring after each, until completely melted and smooth. Alternatively, you can use a double boiler. Set aside to cool slightly.
Step 3: In a separate medium bowl, whisk together the granulated sugar and brown sugar. Add the slightly cooled chocolate mixture to the sugars and stir until well combined.
Step 4: Whisk in the eggs, one at a time, until just incorporated. Don't overmix! Add the vanilla extract and mix until just combined.
Step 5: In a small bowl, whisk together the all-purpose flour, cocoa powder, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed or by hand until just combined. Be careful not to overmix the batter; a few streaks of flour are acceptable.
Step 6: Pour the batter into your prepared 8x8 inch baking pan, spreading it evenly with a spatula.
Step 7: Bake for 25-30 minutes, or until a toothpick inserted into the center comes out with moist crumbs, but not wet batter. The edges should look set, but the center might still look a little gooey.
Step 8: Remove the pan from the oven and let the brownies cool completely in the pan on a wire rack. Once cooled, lift them out using the parchment paper overhang and cut into 16 squares.