Preheat oven to 375 degrees F. In a large bowl beat butter with an electric mixer on medium speed for 30 seconds. Add brown sugar, baking soda, cinnamon, ginger, and salt; beat until combined. Beat in egg, applesauce, and vanilla. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour, the carrots, raisins, and walnuts just until combined.
Drop by slightly rounded teaspoons 2 inches apart onto ungreased cookie sheets. Bake for 8 to 9 minutes or until edges are firm. Transfer to a wire rack; cool.