In a large bowl, cream butter and sugar. Add eggs, 1 at a time, beating well after each addition. Beat in vanilla. Combine dry ingredients; add to creamed mixture alternately with applesauce.
Fill greased or paper-lined muffin cups two-thirds full. Bake at 350° for 25 minutes or until a toothpick inserted in the center comes out clean. Cool for 10 minutes before removing to a wire rack. Frost cooled cupcakes.