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For the Cake: Position rack in middle of oven. Preheat the oven to 350°F/180°C. Coat two 9 x 2-inch (23 cm x 5 cm) round cake pans with nonstick spray, line bottoms with parchment rounds, then spray parchment.
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Whisk the flour, baking powder, baking soda and salt together in a large mixing bowl to aerate and combine; set aside.
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Whisk together oil, sugar and brown sugar until well blended in a mixing bowl. Whisk in eggs one at a time until mixture is smooth and combined. Whisk in cinnamon and vanilla, then use a large, sturdy spatula to fold in the carrots, raisins and nuts.
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Pour wet ingredients over dry and fold in until combined. Divide batter evenly in prepared pans and smooth batter with an offset spatula.
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Bake for about 30 to 35 minutes or until a toothpick shows a few moist crumbs. Cool pans on racks for 10 minutes. Unmold, peel off parchment, and place directly on racks to cool completely. Layers are ready to fill and frost. Alternatively, place layers on cardboards and double wrap in plastic wrap; store at room temperature and assemble within 24 hours.
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Make the Frosting: Beat the butter in a large bowl, using an electric mixer on medium-high speed, until very creamy and smooth, for about 2 minutes. Add the confectioners’ sugar, lemon juice and cream cheese and begin by beating on low speed. As the frosting comes together and you aren’t in danger of confectioners’ sugar flying everywhere, turn speed up to high and beat until very creamy and smooth. Scrape down the bowl once or twice during the process to make sure everything is blended evenly and smoothly. The frosting is ready to use and best if used immediately.
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Apply the Frosting: Decide which frosting application you are going to use. For either approach, place one cake round on serving platter. Spread frosting on top of this bottom cake layer, going all the way to the edges. Place second layer on top.
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If you want to leave the cake sides bare, apply the remaining frosting just to the top of the uppermost layer, leaving the sides naked and pristine.
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If you want to schmear the sides with a thin veil of frosting, simply do just that! I like to use a small straight icing spatula. Apply a thin layer to the sides and be a little more generous for the top.
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The cake is done and needs no additional décor. If you want to replicate the images, simply by some slender fresh carrots with their tops attached and trim the carrots themselves and then press them into the frosting right before serving. Some extra cinnamon can be sprinkled on top to look like dirt, it you like.
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The cake may be served immediately, or refrigerated for up to 2 days under a cake dome (so as to not disturb the frosting). Bring to room temperature before serving.