In a mixing bowl, cream butter and sugar. Add the eggs, one at a time, beating well after each addition. Beat in applesauce and vanilla. Combine flour, baking powder, salt, cinnamon and nutmeg; add to creamed mixture alternating with the milk. (The batter will be thick). In a skillet or deep fryer, heat oil to 375. Drop by teaspoonfuls, a few at a time into hot oil. Turn with a slotted spoon and fry until golden brown, about 1 minute on each side. Drain on a paper towel and Roll in powder sugar while warm. Yield 5 dozen.