Preheat oven to 350 degrees. Grease 2 (2 1/2-inch) wide-mouth canning jars with a paper towel dipped in shortening.
Sift flour, baking powder, salt, nutmeg, allspice, cinnamon, and baking soda into a large bowl. Using the electric mixer and a medium mixing bowl, cream together the shortening and sugar on low speed, then increase to medium mixing speed to mix well. While the mixer is running, add the eggs, one at a time. Stop the mixer and scrape down the sides. Remove the bowl from the mixer and stir in the applesauce. With a large spoon, stir in the sifted flour mixture and the walnuts. Pour the batter into a measuring cup with a spout; pour the batter neatly into greased jars. Fill the jars halfway full. Place the filled jars on the sheet tray and bake on the center rack of the oven for 45 minutes. When done, the top of the cake will spring back when pressed with a finger, and the sides will have come away from the jar. Remove from oven and let cool 10 to 15 minutes. Cake will still be warm, but not hot. Place a circle of waxed paper on top of the jar and seal with the jar lid. Place carefully so as not to break the glass.