1. For Cake: Preheat oven to 350F. Butter two 8-inch round cake pans, line with parchment paper and butter paper. Dust pan with flour.
2. Toss carrots, walnuts and 1/2 cup pineapple with 1/2 cup flour in a small bowl and set aside.
3. Whisk remaining 1 1/2 cups flour, cinnamon, nutmeg, ginger, baking soda, baking powder and salt together in a medium bowl.
4. Beat eggs and sugar in a large bowl using a mixer until thick and light, about 5 minutes. Gradually add oil, while beating. Beat in pureed carrot.
5. Fold in dry ingredients until a loose batter forms. Fold in nuts, carrots, and pineapple. Pour into prepared pans. Bake until firm to touch and a cake tester, inserted into center, comes out clean, about 45 minutes. Cool cakes in pans on a rack for 25 minutes. Turn out of pans and cool completely on rack.
6. For Icing: Beat cream cheese in a large bowl, using a mixer until smooth and fluffy. Gradually beat in butter until smooth. Sift sugar over cream cheese mixture; beat until smooth. Add lemon zest and vanilla extract and beat until light and fluffy. Refrigerate icing for about 20 minutes.
7. Place a cake layer on a cake stand or plate. Spread about half icing over top. Sprinkle on half of pineapple and top with remaining cake layer. Spread remaining icing over top and sprinkle with remaining pineapple. Garnish with thin peels of fresh carrot.
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