Preheat oven to 350°F (175°C) and grease two 8-inch round cake pans.
Prepare flax eggs by mixing flaxseed meal with water; set aside for 5 minutes to thicken.
In a large mixing bowl, sift together flour, sugar, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg.
In a separate bowl, whisk together the vegetable oil, applesauce, 2 tsp vanilla extract, and prepared flax eggs.
Gently fold the wet ingredients into the dry ingredients until no flour streaks remain.
Fold in the grated carrots and chopped walnuts.
Distribute batter evenly into the prepared pans and bake for 30 to 35 minutes until a toothpick inserted in the center comes out clean.
Allow cakes to cool in the pans for 10 minutes, then transfer to a wire rack to cool completely.
For the frosting, beat vegan cream cheese and vegan butter together until smooth.
Slowly incorporate powdered sugar and 1 tsp vanilla extract, beating on high until light and fluffy.
Apply a generous layer of frosting between the cooled cake layers and across the top and sides.