In a large bowl, cream together the butter, shortening, white sugar and brown sugar until smooth. Beat in the eggs one at a time, then stir in the vanilla. Combine the all-purpose flour, whole wheat flour, oats, baking powder and baking soda; blend into the creamed mixture to form a dough. Fold in the white chocolate chips and pistachios. Drop dough by teaspoonfuls onto ungreased cookie sheets.
Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.
