Preheat oven to 350°.
Beat sugar and butter at medium speed of a mixer until well-blended (about 5 minutes). Add applesauce and egg; beat well. Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, oats, soda, cinnamon, and salt. Add flour mixture to sugar mixture; beat at low speed until well-blended. Stir in raisins, walnuts, and vanilla. Using 3 level tablespoons per cookie, drop dough 2 inches apart onto baking sheets coated with cooking spray; flatten slightly. Bake at 350° for 10 to 15 minutes. Remove cookies from pans; cool on wire racks.