Preheat the oven to 350°F (180°C). Line a cupcake tray with cupcake liners. Set aside. 

Sift all purpose flour into a mixing bowl and add light brown sugar, baking soda, baking powder, salt, cinnamon and nutmeg. Mix together. 

Prepare your flax eggs by adding 2 tablespoons of ground flaxseed meal to a bowl and adding in 6 tablespoons of hot water (from the kettle). Let it sit for a minute to become gloopy. 

Add the flax eggs, grated carrot, oil, applesauce, vanilla extract and vinegar and mix into a thick batter. 

If the batter seems too thick to come together then step away from the bowl for 2-3 minutes and then come back to it and mix again. The moisture from the freshly grated carrot will seep into the batter, so that when you mix it again it will come together into a batter. 

Divide the batter between the 12 cupcake liners. 

Bake for 25 minutes. Insert a toothpick into the center of one of the cupcakes, it should come out clean. 

Let the cupcakes cool for a few minutes and then remove them from the cupcake pan and transfer them to a wire cooling rack to cool completely before frosting. 

Prepare your frosting by adding the powdered sugar, vegan butter, vanilla extract, apple cider vinegar and 2 tablespoons of the lemon juice to the bowl of your stand mixer. 

Start at slow speed, gradually increasing speed until the frosting is thick and smooth. If needed, add the extra lemon juice. 

When the cupcakes are completely cool, pipe on the frosting and decorate with chopped walnuts.