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.