Prepare the Pan: Grease an 8 × 8-inch (20 × 20 cm) square baking pan, then line it with parchment paper, leaving about 2–3 inches of parchment extending over the sides to create handles. Set the prepared pan aside.
Preheat the Oven: Preheat the oven to 350°F (180°C).
Melt the Chocolate Mixture: Place the butter pieces, Dutch-processed cocoa powder, and vanilla in a microwave-safe bowl. Heat in 10-second increments, stirring thoroughly between each interval, until the butter and cocoa mixture is completely melted and combined. Alternatively, melt the mixture in a saucepan over simmering water over medium heat, whisking continuously. Set aside to cool. Combine the Dry Ingredients: In a large bowl, whisk together the all-purpose flour and salt until evenly combined. Set aside.
Beat the Eggs and Sugar: Place the eggs and sugar together in another large bowl. Using a hand mixer or stand mixer, beat on high speed for about 5–8 minutes, until the mixture becomes thick and resembles cake batter.
Combine the Wet Mixtures: Once the melted cocoa mixture has cooled, add it to the beaten egg-and-sugar mixture. Beat until the ingredients are thoroughly combined.
Fold in the Dry Ingredients: Add the flour mixture to the wet ingredients. Gently fold with a silicone spatula until no white streaks remain. Fold in the chopped semi-sweet chocolate if using. Avoid overmixing.
Fill the Pan: Transfer the brownie batter to the prepared pan and spread it evenly across the surface.
Bake the Brownies: Bake at 350°F (180°C) for about 30 minutes. Begin checking at 28 minutes, then check every 2 minutes afterward. The brownies are ready when a toothpick inserted into the center comes out with a few moist crumbs. Avoid overbaking to preserve the fudgy texture. Cool in the Pan: Allow the brownies to cool in the pan for at least 20 minutes.
Cool Completely: Use the parchment-paper overhang to carefully lift the brownies from the pan and transfer them to a wire rack. Allow them to cool completely.
Slice and Serve: Cut the cooled brownie slab into squares and serve.