Preheat the oven to 350F. Butter a 9-inch square pan. Set aside.
Sift together the flour, cocoa powder, and salt; set aside
Melt half the butter in a medium bowl in the microwave. Add the remaining butter to the melted butter and stir. The butter should look creamy, with small bits of unmelted butter, and be at room temperature.
In a bowl of a stand mixer fitted with the paddle, mix together the eggs and sugar on medium speed for about 3 minutes, or until thick and very pale. Mix in the vanilla. On low speed, add about one-third of the dry ingredients, then add one-third of the butter, and continue alternating the remaining flour and butter. Add the chocolate and mix to combine.
Spread the batter evenly in the pan. Bake for 40 to 45 minutes, until a cake tester or wooden skewer poked into the center comes out with just a few moist crumbs sticking to it. If the pick comes out wet, test a second time, because you may have hit a piece of chocolate chip; then bake for a few more minutes longer if necessary.
Cool in the pan until the brownie is at room temperature. Run a knife around the edges, and invert the brownie onto a cutting board. Cut into 12 rectangles or 16 squares. Dust the tops with powdered sugar just before serving.
The brownies can be stored in an airtight container for up to 3 days.