Butter and flour an (8-inch square) baking pan and preheat the oven to 350 degrees F.
In a small heavy saucepan melt the chocolate and butter over low heat, stirring until the mixture is smooth, and let the mixture cool completely. Into a bowl sift together the flour, baking powder, and salt. In a large bowl, using an electric mixer, beat the eggs. While still beating, add the sugar, a little at a time, and beat the mixture at high speed for 3 minutes, or until thick and pale.
Add in the chocolate mixture and the vanilla, and then add in the flour mixture, stirring until the mixture is blended well. Stir in the walnuts. Pour the batter into the baking pan, smoothing the top. Bake the brownies in the middle of the oven for 25 to 30 minutes, or until the brownie pulls away slightly from the sides of the pan, and a cake tester, inserted in the center, comes out with crumbs adhering to it. Let the brownies cool completely in the pan before cutting them into squares.