Step 1Heat the oven to 325 degrees. Place the walnuts on a jellyroll pan and bake until they taste toasty, 10 to 15 minutes. Check them by eating one. To remove the brown skin, put the baked nuts in a sieve or strainer and shake them. The brown stuff will fall off. Alternatively, you can rub them in a towel.
Step 2Sift together the flour, baking soda, salt and cinnamon. In a mixing bowl, blend the beaten eggs, white sugar, brown sugar, vanilla and corn oil with one-fourth cup water until the ingredients are just combined. Then stir in the flour mixture and toasted nuts. When thoroughly blended, add the apples and stir.
Step 3Pour the batter into a greased 9-by-13-inch baking pan and bake until a toothpick inserted in the center comes out dry, 60 to 70 minutes. The cake will appear moist but not wet. Cut the cake for serving and sprinkle each slice with powdered sugar.