Preheat the oven to 350F (180C). Line the bottom of the cake pans with baking paper and lightly grease the sides. Alternatively, lightly grease and flour the bottom and sides of the cake pans.
Place the chopped walnuts on a baking tray and bake in the oven for about 6-8 minutes, or until you start to smell a nutty aroma. Remove from the oven and set aside.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg and salt. Set aside.
In a large bowl (this recipe has a lot of batter), beat together the granulated sugar, brown sugar and eggs until frothy.
Then beat the oil, vanilla and applesauce into the sugar and egg mixture.
Beat the flour mixture into the wet ingredients, starting with the mixer on a low speed and gradually increasing until combined.
Turn off the mixer and stir in the grated carrots and chopped nuts.
Divide the batter evenly between the cake pans. Bake in the middle of the oven (ensure both cake pans are on the same rack) for about 34-38 minutes, or until an inserted toothpick comes out clean. If using two 9-inch cake pans, the bake time will be about 30-35 minutes. If you are unsure if the cake is done, I recommend baking for a few extra minutes because this cake is so moist and unlikely to dry out.
Cool the cakes in the cake pans for at least 15 minutes, then carefully transfer to a wire rack to continue cooling.