Preheat oven to 350℉/180°C, then grease and flour two 9-inch cake pans, tapping out any excess flour. (Alternatively, you can grease the bottom of the pans, then line them with parchment paper for even easier cake removal.)
In a large mixing bowl, whisk together the flour, baking soda, baking powder, salt and spices.
Place the coconut oil in a microwave-safe bowl. Melt the oil in the microwave by heating in 15-second bursts, then stir in the sugar, applesauce and the vanilla extract until thoroughly combined, at least 2-3 minutes.
Pour the flour mixture into the wet mixture, then fold in the shredded carrots, and nuts.
Divide the cake batter into the two prepared pans. Bake for 35 minutes, or until a toothpick inserted into the centre comes out clean or almost clean (a few crumbs are fine).
Place the cake pans on wire racks and let them cool for 20 minutes. After 20 minutes, carefully remove the cakes from the pans and let them finish cooling on the wire racks.
To make the frosting and assemble:
Beat the butter and cream cheese for about 2 minutes, or until fully incorporated. Beat in the vanilla extract and salt, followed by the sifted cornstarch and powdered sugar. Once the first cup is incorporated, add the next, then the next. If the frosting has reached your desired sweetness and consistency, stop there. If not, add the fourth cup of sugar. I usually add all 4 cups to reach a stiffness that I can pipe.
Refrigerate the bowl of frosting for at least 15 minutes before using.
Set one of the cake layers on a cake plate or lazy Susan. Frost the top of the first layer and set the second layer on top. Then, frost the sides and top of the cake. Garnish with nuts or decorate any way you like and refrigerate for at least 15 minutes before serving to set the frosting.
If storing in the fridge, wrap the cake in saran wrap.
Notes
Storage instructions: Store your vegan carrot cake in the refrigerator for up to 4 days; I recommend wrapping it in plastic wrap or transferring it to an airtight storage container if you have leftovers. Let the cake sit on the countertop for 15 minutes or so before serving.Freezing instructions: If you want to freeze the whole cake, place it on a parchment-lined baking sheet and freeze it uncovered. Once the frosting is solid, you can wrap the cake in a few layers of plastic wrap and return it to the freezer for up to 3 months. Leftovers can be frozen in an airtight storage container.