Preheat the oven to 180°C.
Make the flax eggs: mix 2 Tbsp ground flaxseed with 6 Tbsp of warm water and set aside.
Grease two 20 cm round cake tins and line the base with parchment paper.
In a large mixing bowl, combine the flour, baking powder, baking soda and cinnamon. Add the sugar, mix well.
Mix the oil, vanilla, vinegar and flax eggs in another bowl, then pour this mixture into the batter together with the grated carrot. Combine well, adding the apple juice as you’re stirring the batter. Rest for 5 minutes.
If the batter is too thick, stir in some almond milk. Lastly, fold in the chopped walnuts if used. The batter should not be dense but not runny either.
Divide the mixture between the cake tins, spreading it out evenly.
Bake in the middle of the oven for about 25 minutes or until a skewer inserted into the centre comes out clean.
Transfer the cakes to a wire rack and let them cool completely.
Meanwhile, prepare your frosting: mix the vegan cream cheese with lemon juice, almond milk, sugar, vanilla and maple syrup. Adjust it to your liking (it should be spreadable but not too thin!).
Spread some frosting on top of one cake, then put the other one on top and frost the cake all over (top and sides).
Sprinkle some chopped walnuts on top of the cake and it's ready to serve.