Preheat oven to 350 degrees F. Grease a 9-inch square baking pan and line with parchment paper or foil (for easy removal, if desired).
In a medium bowl, stir together the butter, sugar and vanilla with a wooden spoon. Add the eggs and beat well with the spoon. Add the flour, cocoa powder, baking powder and salt and stir gently until well blended. Stir in the nuts, if using. Spread the batter evenly into the prepared pan.
Bake for 20 to 25 minutes, or until brownies begin to pull away from sides of the pan and a toothpick inserted in the center comes out with moist crumbs attached. Cool completely in the a pan on wire rack. Cut into squares and serve. (If you used parchment paper or foil, use that to lift the brownies out and cut on a cutting board.) The brownies can be stored in an airtight container at room temperature for up to 4 days.
These brownies are on the thin side; if you like thicker brownies, bake them in an 8x8-inch square pan and increase the baking time by a couple of minutes.