In the top of a double boiler, melt together the butter and chocolate over very low heat. Remove the inner pan and allow the mixture to partially cool, then stir it until smooth.
In a bowl, beat the sugar and eggs with an electric mixer at high speed until light and airy, about 4 to 5 minutes. Blend in the vanilla extract and partially cooled chocolate. Chop the walnuts very fine (a food processor works well for this) and stir them into the chocolate mixture.
Sift the flour, baking powder, and salt into a medium bowl. Stir the dry ingredients into the chocolate mixture half at a time, until the dough is evenly mixed – it will be quite soft. Cover and refrigerate the dough for 3 to 4 hours.
Heat the oven to 325 degrees. Grease 2 large cookie sheets or line them with parchment paper. Put about 1 cup of Confectioners’ sugar in a small bowl. Now shape the dough into 1-1/2-inch-thick balls. Generously coat each one with confectioners’ sugar and place it on a baking sheet, leaving about 3 inches between cookies. Bake the cookies on the center rack, 1 sheet at a time, for about 13 minutes. When done, the cookies will have puffed and will feel very soft to the touch. Do not overbake.
Cool the cookies on the sheet for 15 minutes, then transfer them to a wire rack and let them cool thoroughly. TIP: To lengthen the shelf life of these cookies, refrigerate them for 1 to 2 hours, then wrap them individually in plastic wrap. Makes 24 to 30 cookies.