Position a rack in the center of the oven and preheat to 350ºF. Butter a 9-inch round cake pan and line the bottom with a round of parchment paper cut to fit.

In a large bowl, stir together the grated carrots, eggs, sugar, and oil until combined. Place a medium mesh strainer over the bowl and add the oat, sweet rice, millet, and tapioca flours along with the baking powder, baking soda, salt, cinnamon, and ginger. Sift the dry ingredients into the wet ingredients, stirring until combined. Stir in the walnuts.

Scrape the batter into the prepared pan and smooth into an even layer. Bake the cake until a toothpick inserted near the center comes out clean, 22-30 minutes. Remove from the oven and let cool 20 minutes in the pan. Turn the cake out onto a serving board to cool completely.

Spread the cream cheese frosting over the cooled cake. Chill briefly if the frosting is becoming overly soft.

Peel the carrots. Lay a carrot on a cutting board and begin peeling long, wide strips from one side. Repeat with the other carrots; you’ll have more carrot ribbons than you need for one cake. (Throw them in a salad!) Use your fingers to curl the ribbons into rounds and stick them into the frosting; the frosting will hold them in place. Sprinkle the cake with finely chopped walnuts and decorate with carrot greens.

Chill the cake until firm enough to slice, 30 minutes and up to several hours, then cut into wedges. Let the cake slices stand at room temperature to soften the frosting and serve. Leftover cake keeps well, refrigerated airtight, for up to 4 days.