Pre-heat the oven to 350°F and place the oven rack in the center of the oven.
In a medium bowl, sift together the flour, baking soda, baking powder, cinnamon, nutmeg and cloves. Set aside.
Dump the walnuts onto a cutting board and mince into very tiny pieces; you can also use a food processor or chopper. Set aside.
In a large bowl using an electric mixer on medium speed, cream the butter until smooth. Add the sugar and mix on medium speed until fluffy. On low speed, mix in the egg and vanilla, until just combined. Add in the walnuts and mix at low speed until combined. Next, add the applesauce and the flour mixture a third of each at a time, beginning with the applesauce and alternating to end with the dry ingredients. Mix each addition at low speed until just combined. Do not over mix, or the batter will become tough.
If you are not using nonstick bakeware, grease your bakeware with butter and flour it.
For cake, pour the batter into a 9-inch x 13-inch baking pan and bake for 35-40 minutes, until the top springs when lightly touched and a knife inserted in the center comes out clean. For muffins, pour the batter into a 12-cup muffin pan and bake for 25-30 minutes.