Preheat oven up to 350°F. Place cupcake wrappers in cupcake pan, two 8-inch cake pans, or spray 9x13 pan with nonstick spray. Set aside.
In a large bowl, whisk together the brown sugar, granulated sugar, oil, eggs, applesauce, and vanilla until combined. Set aside. In another large bowl, whisk the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves together until combined. Pour the dry ingredients into the wet ingredients and fold together until just combined and no streaks of flour are left. Fold in the carrots and pineapple.
Spoon the batter into the prepared pans or, using an ice cream scoop, scoop about ¼ cup batter into each cupcake wrapper. Bake 9x13 pan for 38-42 minutes, two 8-inch pans for 38-44 minutes or cupcakes for 18-20 minutes or until a toothpick inserted into the center comes out clean. Allow to cool at least 10 minutes then remove from pans to cool on wire rack.
Make the frosting: In the bowl of a stand mixer or using a handheld mixer fitted with a whisk attachment, beat the cream cheese, butter, vanilla, salt and 2 cups of the icing sugar together on medium speed until smooth, 2 to 3 minutes. Add the remaining icing sugar and beat for 2 minutes. If the icing is very soft, refrigerate it for about 30 minutes until is firmed up a bit and at a piping for frosting consistency.
You can either pipe or spread the icing on the cake or cupcakes once they are completely cool.
Cover tightly and store at room temperature or in the refrigerator for up to 4 days. This cake is great to make the day before it is needed. Prepare the cake/cupcakes and the frosting 1 day in advance. Store cake at room temperature, covered tightly. Refrigerate prepared frosting in an airtight container until ready to use. Bring icing to room temperature then frost cake.