In a large bowl, cream butter and sugar until light and fluffy. 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; gradually add to the creamed mixture alternately with milk (the batter will be thick).
In an electric skillet or deep-fat fryer, heat oil to 375°. Drop teaspoonfuls of batter a few at a time into hot oil. Fry until golden brown on both sides. Drain on paper towels; roll in sugar while warm.