Preheat your oven to 350°F (180°C). It's crucial to get this right! While that’s happening, take your two 7- or 8-inch cake pans and spray them lightly with a non-stick cooking spray. For added insurance against sticking, cut out circles of parchment paper to line the bottoms of the pans. This little extra step ensures your cakes slide out effortlessly.

In a large mixing bowl, sift together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and nutmeg. Whisk them well until they are combined. This helps distribute the leavening agents evenly and will make for a light and fluffy cake.

Add the light brown sugar and grated carrots directly to the bowl with the dry ingredients. Take a moment to appreciate the bright orange color of the shredded carrots—it's a good sign of moisture and flavor!

Now, let’s get those flax eggs ready. In a small bowl, combine the ground flaxseed meal and hot water. Allow it to sit for about a minute. Don’t worry if it looks a little gloopy; that's what we want! Flaxseed serves as a great binder here, much like eggs would in traditional cakes.

Once your flax eggs have set, add them to the mixing bowl along with the canola oil, vanilla extract, apple cider vinegar, and the optional chopped walnuts. The vinegar is essential as it reacts with the baking soda to help the cake rise.

Now, mix everything together with a spatula or a wooden spoon. The mixture may seem a bit dry at first, but don’t panic! Step away for a couple of minutes. The carrots will release some moisture, so the batter should come together into a thick yet smooth consistency.

Once the batter is ready, evenly divide it between the two prepared cake pans. Use a spatula to smooth the tops. Bake for about 30 minutes. To check for doneness, insert a toothpick into the center of one of the cakes. If it comes out clean, it’s ready!

When the cakes are baked, let them cool in the pans for about 10 minutes. This makes them easier to remove. Then, gently transfer them to a wire cooling rack to cool completely. This is where the aroma really fills your kitchen—trust me, you won’t want to rush this part!

While your cakes are cooling, it’s time for the lemon buttercream frosting. In the bowl of a stand mixer, combine the powdered sugar, vegan butter, vanilla extract, apple cider vinegar, and 2 tablespoons of lemon juice. In a rush? You can whisk it all by hand, but a stand mixer makes things easier. Start on low speed, gradually increasing until your frosting is thick and smooth.

If your frosting seems too thick, don't fret! Add in a little more lemon juice, one drop at a time, until you get a spreadable consistency. Taste it, and you may just want to dance with joy!

Once your cakes are completely cool, it’s time to frost. Place one layer on a serving platter and spread a generous amount of frosting on top. Add the second layer, then spread more frosting on the top and sides. Finish by sprinkling chopped walnuts on top for that nice crunch and aesthetic touch.