Make the cake: Preheat the oven to 350°F. Grease three 9-inch cake pans, line with parchment paper, then great the parchment paper as well.
Whisk together the brown sugar, granulated sugar, applesauce, eggs, and vanilla in a large bowl until combined and no lumps remain. In another large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger, cloves, allspice, and nutmeg. Pour the wet ingredients into the dry ingredients and using a rubber spatula, fold the ingredients together until just combined. Fold in the carrots and the walnuts.
Pour the batter evenly into the cake pans— I normally use a dry measuring cup to scoop. Bake for 20-22 minutes. Test the center with a toothpick. if it comes out clean, the cakes are done. If not, bake until cooked through in 2-3 minute increments. Allow the cakes to cool completely in the pans on a wire rack. The cakes must be completely cooled before frosting.
Make the frosting: In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the cream cheese and butter together on high until smooth and creamy. Add the heavy cream, vanilla, and salt. Then, turning the mixer to low, slowly add the powdered sugar. Then switch back to high speed and beat for 2 minutes. Scrape down the sides and bottom, then beat together for another 30 seconds.
Assemble and frost: Using a large serrated knife or cake leveler, layer off the tops of the cakes to create a flat surface. Place 1 cake layer on your cake stand and evenly cover the top with frosting. Top with 2nd layer, more frosting, and then top with the 3rd layer. Spread remaining frosting all over the top and sides. Decorate the top of the cake with the remaining walnuts. Refrigerate cake for at least 20-30 minutes before slicing. This helps the cake hold its shape when cutting. Cover the leftover frosted cake tightly and store in the refrigerator for up to 5 days.