Preheat your oven to 350°F (175°C). Grease an 8x8-inch baking pan with butter or non-stick cooking spray and line it with parchment paper, leaving an overhang on two sides for easy removal.
Combine the softened butter and chopped unsweetened chocolate in a medium microwave-safe bowl. Microwave in 30-second intervals, stirring after each, until completely melted and smooth. Let the mixture cool for 5 minutes.
Whisk the granulated sugar into the cooled chocolate mixture until well combined. Then whisk in the eggs, one at a time, followed by the vanilla extract, until the batter is smooth and glossy.
Sift the flour, cocoa powder, and salt directly into the wet ingredients. Use a spatula to fold the dry ingredients into the wet until just combined and no streaks of flour remain, being careful not to overmix. Gently fold in the chocolate chips.
Pour the thick batter into the prepared pan and use the spatula to spread it into an even layer. Bake for 25-30 minutes, or until the top is set and has a crackly appearance and a toothpick inserted into the center comes out with a few moist crumbs attached.
Let the brownies cool completely in the pan set on a wire rack for at least 1 hour before using the parchment paper overhang to lift them out. Cut into 16 squares and dust with powdered sugar just before serving.