Preheat the oven to 350°F (180°C). Spray two 7-inch cake pans or two 8-inch cake pans (see notes*) with non-stick spray and line the bottoms with circles of parchment paper.
Sift the all purpose flour into a mixing bowl and add the baking powder, baking soda, salt, cinnamon and nutmeg. Mix together.
Add the light brown sugar and grated carrot to the mixing bowl.
Prepare your flax eggs by adding 2 tablespoons of ground flaxseed meal to a bowl with 6 tablespoons of hot water. Let it sit for a minute to become gloopy.
Add the flax eggs, oil, vanilla extract, apple cider vinegar and chopped walnuts to the mixing bowl.
Mix everything together. If it seems like it won't come together and the mix is too dry then step away from the bowl for a couple of minutes and then come back to it. The carrots will have released more moisture into the batter allowing it to come together into a thick batter.
Divide the batter evenly between the two prepared cake pans.
Bake for 30 minutes or until a toothpick inserted into the center of one of the cakes comes out clean.
Allow the cakes to cool in the pans for a few minutes before transferring them to a wire cooling rack to cool completely.
While the cakes are cooling, prepare your frosting.
Add the powdered sugar, vegan butter, vanilla extract, apple cider vinegar and 2 tablespoons of lemon juice to the bowl of your stand mixer.
Start mixing on low speed gradually increasing speed until your frosting is thick and smooth.
If your frosting is too thick then add in the extra tablespoon of lemon juice, a drop at a time and only as much as needed to get to a spreadable consistency.
Frost the cake and decorate the top with chopped walnuts.