Step 1In a small saucepan over medium heat, bring the carrot juice to a boil, then reduce the heat to low. Slowly reduce the juice, stirring occasionally, until it is reduced to one-fourth cup and is darkened in color, about 20 minutes. Remove from heat and set aside to cool.
Step 2Heat the oven to 350 degrees. Cut a circle of parchment paper to fit the base of a 9-inch cake pan. Grease both sides of the paper with oil and place it in the pan. Set aside.
Step 3Into a large bowl, sift together the almond flour, tapioca flour, sweet rice flour and brown rice flour. Sift in the xanthan gum, guar gum, baking powder, salt, cinnamon, nutmeg, one-half teaspoon ginger and cloves. Stir together the flour-and-spice mixture to make sure the ingredients are evenly distributed and set aside.
Step 4In the bowl of a stand mixer, or in a large bowl using a hand mixer, beat 1 cup butter and the white and brown sugars until creamy and fluffy, 3 to 5 minutes. Add the 4 whole eggs and 2 egg yolks, one at a time, until fully incorporated. Beat in the orange zest and vanilla, then the reduced carrot juice.
Step 5Slowly add the flour-and-spice mixture to the liquid until combined. Gently stir in the carrots, and walnuts and raisins (if using). The cake batter should be quite thick, but you should still be able to work a rubber spatula through it easily.
Step 6In a clean, dry mixing bowl, beat the 2 remaining egg whites with the cream of tartar with the hand mixer until they form stiff peaks. Gently fold the stiff egg whites into the cake batter until they are entirely incorporated.
Step 7Pour the batter into the prepared cake pan. Slide the pan into the oven and bake until a knife inserted into the center of the cake comes out clean and the top is golden brown, about 45 minutes. Transfer the pan to a wire rack and set aside for 15 minutes before removing it from the pan. Turn the cake out onto the rack, turn it right side up, and cool completely before frosting it.
Step 8Make the frosting: In a mixing bowl, beat together the cream cheese, remaining 1 cup butter, remaining 1 tablespoon ginger and cardamom with the hand mixer until smooth. Gently beat in 3 cups powdered sugar, then beat in more if you prefer the frosting thicker and sweeter. Spread the frosting on the top and sides of the cake before slicing and serving.