Preheat oven to 350°F (175°C). Line a 9" x 5" loaf pan with parchment paper so that it overhangs by a couple of inches on the two long sides.
Place the butter and unsweetened chocolate in a large microwave-safe bowl. Heat at half-power in 30-second increments until the butter is melted and the chocolate melts when stirred. Stir until smooth. Allow to cool slightly.
Whisk in the cocoa powder and sugar. Then, mix in the egg and vanilla.
Add the flour and salt, and stir just until combined.
Transfer the batter to the prepared pan, and spread evenly.
Bake 22 to 26 minutes, or until a pick inserted into the center comes out with a few moist crumbs attached.
Allow the brownies to cool completely in the pan on a wire rack. Use the overhanging parchment paper to lift the brownies from the pan to a cutting board. Cut into bars.