Preheat oven to 350°F. Combine chopped chocolate and butter pieces in a medium microwavable bowl. Microwave on high until melted and smooth, about 1 minute, stirring after 30 seconds. Set aside to let cool.
Combine eggs and sugar in bowl of an electric mixer and beat on medium-high until pale and fluffy, about 3 minutes. Add melted chocolate mixture; beat on low until smooth. Whisk together flour, cocoa, fine sea salt, and baking powder in a medium bowl. Gradually add flour mixture to batter; beating on low until just combined, about 30 seconds. Spoon batter into a lightly buttered 8-inch square baking dish, and sprinkle with flaky sea salt.