Preheat the oven to 180°C (350°F). Grease two 8-inches round cake pans with butter or cooking spray. Line the bottoms with parchment paper so the cakes come out easily.

In a large bowl, sift together the flour, baking powder, baking soda, salt, cinnamon, ginger, cloves, and nutmeg. Set this mixture aside.

In another large bowl, put the softened butter, brown sugar, and granulated sugar. Use an electric mixer to beat them until the mixture is light and fluffy, about 3–4 minutes.

Beat in the eggs, one at a time. Then add the vegetable oil, applesauce, vanilla extract, and buttermilk. Mix everything until it is well combined.

Gradually add the dry ingredient mixture to the wet ingredients. Stir gently until just combined. Do not over mix.

Add the grated carrots to the batter and fold them in carefully.

Divide the batter evenly between the two prepared pans. Bake in the preheated oven for 40–50 minutes. Check for doneness by inserting a toothpick into the center; it should come out clean. If you are baking at high altitude, check the cakes 5–10 minutes earlier than the suggested time.

Let the cakes cool in the pans for 10–15 minutes. Then remove them from the pans and transfer to a cooling rack. Allow them to cool completely before frosting.

Beat the butter and cream cheese together until creamy. Add heavy cream, powdered sugar, vanilla, and salt. Beat until smooth and fluffy.

If necessary, level the cakes with a serrated knife. Spread a layer of frosting on top of the first cake layer, place the second layer on top, and frost the top and sides evenly. Chill for 20–30 minutes before slicing for easier cutting.