Preheat the oven to 350°F. Line an 8-inch square pan with foil and butter the foil.
Sift together the flour, cocoa, baking powder, and salt.
Melt the butter in a 3-quart saucepan over moderate heat. Add the sugar and vanilla and stir to combine. Remove the pan from the heat and let cool for 5 minutes.
Crack the eggs into a small bowl and then add the eggs to the saucepan, stirring rapidly to avoid cooking them.
Add the sifted dry ingredients and walnuts to the saucepan and stir.
Spread the mixture smoothly in the prepared baking pan and bake for 20 to 25 minutes, or until a toothpick inserted in the center just barely comes out clean. Let cool in the pan for 15 minutes.
Cover the pan with a cooling rack and flip the pan over. Remove the pan and foil lining from the brownies. Then cover the brownies with a cutting board and flip them back over. Remove the cooling rack from the top of the brownies.