Preheat the oven to 350 degrees Fahrenheit. Lightly spray two cake pans with oil.
In a mixing bowl, whisk together all of the dry ingredients until combined.
In a separate bowl, whisk together the aquafaba, applesauce, and soy milk. Add the orange juice, vanilla, and sugar, and stir until well-combined. Add the oil and mix vigorously until the oil is fully incorporated.
Using a spatula, add the dry ingredients to the wet ingredients in 3 batches, gently stirring the mixture together between each batch. Stir until no large pockets of dry ingredients remain; small lumps are okay.
Gently fold in the grated sweet potato and stir just until incorporated. Divide the batter evenly into the two prepared cake pans and gently smooth out the top.
Bake for about 25 to 35 minutes, or until a toothpick inserted into the center comes out clean. For best results (especially if your oven tends to heat unevenly), rotate the positioning of the two cake pans at about the 20 minute mark, but avoid opening the oven until then. I usually make the frosting while my cakes bake.
Place the cake pans on a cooling rack and let cool for an hour or so. You can then gently go around the edges with a sharp knife and then, one by one, carefully invert the pan, tapping on the bottom to release the cooked cake. Let cool completely before frosting.
For the toasted walnut frosting:
Drain the cashew pieces and add them to a blender along with the walnuts. As best you can, blend the nuts into crumbs. Add the yogurt, maple syrup, coconut oil, vanilla, lemon juice, and salt, and blend the mixture until completely smooth. Depending on your type of blender, this may take several minutes, and stopping to scrape down the sides.
Adjust lemon, salt, and maple syrup to taste if needed. Refrigerate the frosting in an airtight container to cool it until ready to frost the cake.
Vegan sweet potato cake assembly:
Once the cake and frosting are cooled, place one of the cake layers upside down on your cake stand (or whatever your serving vessel is), so that the side that was the bottom during baking is facing up (this is the easiest way to get a flat surface for frosting).
Take about half of the frosting and spread it out evenly over the top of the cake. If desired, sprinkle with toasted walnuts.
Take the second cake layer, and place it on top of the frosted bottom layer. Spread the remaining frosting on top, and garnish with toasted walnuts to taste.
Serve cold. Leftover cake will keep in the refrigerator for a couple of days.
Notes
FLOUR MEASUREMENT: If measured by scooping, the flour is approximately 2 cups. If you measure by spoon-and-level method, the flour is approximately 2 and 1/2 cups. For best results, measure by weight.AQUAFABA: Aquafaba is bean cooking liquid. For this recipe, use the liquid from a (preferably BPA-free) can of chickpeas or white beans. If using homemade cooked beans, be sure to use the cooking liquid, NOT the soaking liquid, and if needed, reduce it on the stove until the consistency is slightly viscous.NON-DAIRY YOGURT: Some people have trouble finding good non-dairy yogurt in their area. If this applies to you, just omit it and add another 2 teaspoons of lemon juice, plus a splash of non-dairy milk as needed to blend.FLAT CAKE LAYERS: This cake doesn't tend to puff up a lot, so the layers should be even enough already to frost easily. If for some reason this isn't the case, use a very sharp knife to slice off and excess domed areas. You can then hide those extra cake pieces for yourself and snack on them later :)NUTRITION: Nutrition facts include the frosting but not extra chopped walnuts for topping.