Preheat Oven: Preheat the oven to 180°C (160°C fan).
Prepare the Flax Eggs: In a small bowl or cup, mix the ground flaxseeds with 9 tbsp of water. Stir well and let sit for 5 minutes until thick and gloopy.
Mix Dry Ingredients: In a large bowl, combine the gluten-free flour, baking powder, baking soda, cinnamon, salt, and allspice. Stir in the chopped walnuts.
Combine Wet Ingredients: In a separate bowl, mix together the sunflower oil, grated carrots, maple syrup, vanilla extract, and the prepared flax egg mixture.
Combine Wet and Dry Mixtures: Pour the wet mixture into the dry ingredients, mixing thoroughly until a smooth batter forms.
Prepare the Baking Tray: Line a standard 12-cup muffin tin with cupcake liners. Evenly distribute the batter among the liners. If there is any extra batter, you can bake additional cupcakes or a small cake separately.
Bake: Bake in the preheated oven for about 20 minutes, or until a skewer inserted into the centre of a cupcake comes out clean.
Make the Icing: While the cupcakes are baking, prepare the icing. In a small saucepan, combine the vegan cream cheese, coconut oil, and agave syrup. Gently heat, whisking regularly, until the coconut oil melts. Remove from heat and transfer to a blender or use a hand mixer to blend until emulsified and thick, similar to a mayonnaise-like consistency. Place in the fridge to cool and thicken.
Cool and Ice the Cupcakes: Once the cupcakes are fully cooled, use a piping bag or a spoon to ice each cupcake. Garnish with a walnut on top.