Beat butter or margarine and shortening in a large mixing bowl with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, brown sugar, and baking soda; beat until combined. Beat in cooled chocolate, eggs, and vanilla. Beat in as much of the flour as you can with the mixer. Stir in the remaining flour, chocolate pieces, white baking pieces, and nuts.
Drop dough from rounded teaspoons 2 inches apart on an ungreased cookie sheet. Bake in a 375 degree F oven for 8 to 10 minutes or until tops look dry. Cool on cookie sheet for 1 minute. Transfer to wire racks to cool. Makes about 24.