In a large bowl, cream butter and sugars until light and fluffy. Beat in egg. Beat in applesauce and vanilla. Combine the flour, cracker crumbs, cinnamon, baking soda and allspice; stir into creamed mixture just until moistened. Fold in walnuts and raisins.
Fill greased or paper-lined muffin cups three-fourths full. Bake at 350° for 20-25 minutes or until a toothpick inserted in muffin comes out clean. Cool for 5 minutes before removing from pan to a wire rack. Serve warm.