In a medium saucepan, melt the butter and unsweetened chocolate over medium-low heat, stirring until smooth. Remove from the heat and set aside to cool slightly.
Preheat the oven to 350°F. Grease an 8x8x2-inch baking pan. You can line the pan with foil and grease it for easier removal, but this step is optional.
Stir the sugar into the cooled chocolate mixture. Add the eggs one at a time, mixing with a wooden spoon just until combined. Stir in the vanilla.
In a small bowl, combine the flour and baking soda. Stir the flour mixture into the chocolate mixture just until combined. Spread the batter evenly in the pan.
Bake for 30 minutes, then cool the brownies in the pan on a wire rack.
Frosting
In a small saucepan, melt the chocolate chips over low heat, stirring until smooth. Remove from the heat and let cool.
In a medium bowl, stir together the softened cream cheese and powdered sugar until smooth. Add the melted chocolate and stir with a wooden spoon until fully combined.
Spread the frosting evenly over the cooled brownies.