Preheat the oven to 350°F (180°C). Spray two 8-inch* cake pans with non-stick spray and line the bottoms with parchment paper.
Sift the flour into a mixing bowl and add the brown sugar, baking soda, baking powder, salt, cinnamon and nutmeg and mix together.
Prepare your flax eggs by adding 2 Tablespoons ground flaxseed meal to a bowl and then adding 6 Tablespoons of hot water from the kettle. Leave it to sit for a minute to become gloopy.
Add the flax eggs, applesauce, grated carrot, oil, vanilla extract, apple cider vinegar and chopped walnuts and mix into a thick batter. If it's struggling to come together, just leave the bowl to sit for a couple of minutes and then come back and stir again and it should come together. Leaving it for a minute will allow the grated carrots to release more moisture into the bowl allowing it all to come together.
Divide the batter evenly between the prepared cake tins. Place into the oven and bake for 30-35 minutes or until a toothpick inserted into the center comes out mostly clean. If you have some crumbs on the toothpick, this is fine, but you don't want wet batter.
Let them cool for 5 minutes or so and then remove them from the cake pans and place them onto a wire cooling rack and allow to cool completely before frosting.
Frost with any of the choices of frostings above and decorate with chopped nuts (we used pecans).