Mist an 8-inch, square microwave-safe baking dish with nonstick cooking spray. Place butter and cocoa powder in a large microwave-safe bowl. Microwave on high until butter is melted, 30 to 90 seconds, depending on power and size of your oven (check it every 15 seconds). Add sugar and whisk to combine. Let mixture cool slightly, then whisk in eggs and vanilla. Stir in flour. Scrape batter into prepared baking dish.
Microwave on high until brownies are just set in center, 4 to 7 minutes, depending on power and size of your oven (after 4 minutes, check batter every 15 seconds). Let brownies cool in pan on a wire rack for 10 minutes. Cut brownies into 8 squares and serve.