Preheat the oven to 325°F. Spray two 8-inch round cake pans with cooking spray and line the bottoms with parchment circles.

In the bowl of a stand mixer fitted with the paddle attachment, combine oil, sugar, eggs, and vanilla. Mix on medium speed until well combined, about 2 minutes, occasionally scraping down the bowl.

Add white rice flour, brown rice flour, baking soda, baking powder, xanthan gum, cinnamon, nutmeg, cardamom, and salt to the wet ingredients. Mix on medium speed until smooth, 2–3 minutes. Add carrots and walnuts and mix until all the ingredients are uniformly combined, 1–2 minutes more. Divide the batter evenly between the prepared cake pans.

Bake for 40–45 minutes, rotating the pans between the upper and lower oven racks halfway through baking, until the cake is golden brown and the top feels set when pushed very gently with your finger, or a cake tester or toothpick inserted into the center comes out clean. Cool cakes in pans completely and then unmold onto a wire rack before before trimming and icing.

Using a long serrated knife, carefully trim off the tops of both cakes so the layers are flat on both sides. Place one trimmed cake on a platter facing up, so you’re icing the cut side of the cake. Spread on a thick layer of vanilla bean cream cheese icing. Invert the other layer on top, so the bottom of the second cake is facing up. Spread another layer of icing on the top of the cake. You can either frost the sides or leave them naked, depending on how much icing you personally enjoy and how you want the finished cake to look.