Beat butter at medium speed with an electric mixer until creamy; gradually add sugars, beating well. Add egg and vanilla; beat well. Stir together flour, baking powder, and salt; add to creamed mixture, and beat well. Divide dough into 3 equal portions. Stir 1/2 cup pistachios into 1 portion, 1/3 cup Brazil nuts into another portion, and 1/3 cup walnuts and grated chocolate into third portion. Cover and chill 1 hour.
Divide each portion in half. Keep 1 portion of each dough out to work with, and chill remaining halves. Shape each portion of dough into a 13" x 3/4" log. Place 2 logs, side by side, on a wax paper-lined baking sheet, gently pressing together; place remaining log on top, gently pressing down. Wrap in wax paper. Chill at least 8 hours or freeze until firm. Repeat procedure with remaining dough.
Unwrap dough logs, and slice into 1/4" thick slices; place on greased baking sheets. Bake at 350° for 10 minutes or until lightly browned. Remove to wire racks to cool.
Place chopped chocolate squares in top of a double boiler; bring water to a boil. Reduce heat to low; cook until chocolate melts, stirring occasionally. Leave chocolate over hot water while assembling cookies.
Meanwhile, stir together 1/2 cup pistachios, 1/2 cup Brazil nuts, and 1/2 cup walnuts in a shallow dish. Brush bottoms of cookies with a pastry brush dipped in melted chocolate; sprinkle with nut mixture. Place cookies, nut side up, on wax paper. Let stand 4 to 6 hours or until chocolate is firm.