Get the eggs, as well as the butter and cream cheese for the frosting, out of the refrigerator and allow to come to room temperature.
Brush about a pound of carrots clean. Chop off the yucky parts and discard. Grate the carrots and set aside.
Spray two 9″ cake pans lightly with cooking oil spray. Use round pieces of parchment paper, or cut round pieces, and place them in the bottoms of the pans. Hit the paper lightly with a little more spray.
Weigh or measure the dark brown sugar in the bowl of your stand mixer. Add the vegetable oil to the bowl. Place the bowl in your stand mixer with the paddle attachment and mix on low until combined and smooth.
Add the eggs one at a time. Start each egg on low speed to mix it in, and then increase to medium speed until there’s no sign of it at all. After the last egg, let the mixer run a full minute.
In a separate large bowl, weigh or measure the flour. Add the baking powder, baking soda, salt, cinnamon, nutmeg, allspice, and cloves. Stir with a spoon until it’s all the same color.
With the mixer running at low speed, gradually shake in the flour mixture no more quickly than the mixer can keep it blended. Stop here (and probably once more) to clear the paddle and scrape the sides and bottom of the bowl with your flexible spatula. Now turn the mixer up to medium speed and let it run two full minutes.
Add the pineapple and applesauce, and then run the mixer on low speed long enough to blend.
Remove the paddle, free the bowl from the mixer, and stir in the carrots and optional nuts. Stir until the mixture is well blended.
Pour the batter into the prepared cake pans and place in the preheated oven.It’s a good idea to weigh the two cake pans, and adjust them to the same weight with a spoon, so they get to the same doneness at the same time.
Bake 15 minutes. After 15 minutes, turn the pans 180° in the oven and switch them so they’ll get done evenly.
Bake another 10 to 20 minutes. To check the cakes, first, wiggle the pan and the cake should be firm in the middle. Second, a toothpick inserted deeply into the center should come out clean.Add time in increments of at least two minutes so the oven won’t cool down from opening it.
When the cakes are done, remove them from the oven. Wait about 10 minutes or until the cakes are cool enough to touch.
Use the dull side of a table knife to be sure the sides of the cake aren't stuck to the pan. Place a wire rack on top of each cake pan. Invert the wire rack, cake pan, and all. Thump the top of the cake pan with the heel of your hand to ensure the cake is freed. Remove the cake pan, and then remove the parchment paper. Allow the cakes to cool on the wire rack. Do no proceed with frosting until the cakes are completely cook to the touch.
Make Buttercream Cheese Frosting.
Put one of the layers flat side down on a plate or cake circle. Put four small pieces of wax paper under the layer from four sides so the frosting doesn’t get beyond the cake.Don’t worry about any little grooves or squares from your wire rack.
Scoop enough frosting to separate the layers by half an inch onto the first layer and spread it out with a spatula. Center the second layer flat side down over the first layer, and push down until the frosting reaches the edge. Scoop the rest of the frosting on top of the second layer and spread it out with a spatula. Use a table knife to bring gobs of frosting from the top to the sides of the cake and spread it out. Turn the cake against the knife to smooth the sides. Spread the frosting on top until it is even. Remove the wax paper.
Cake is ready to serve, or if you prefer, you can serve it chilled.