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Preheat oven to 350°F. Grease and lightly flour two 8-inch round cake pans.
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In a blender or food processor, purée chickpeas with lemon juice, oil and lemon zest. Add egg yolks and blend well. Turn mixture into a large bowl.
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In a medium bowl, combine flour, 1⁄2 cup sugar, baking powder, and salt. Mix well. Add to puréed chickpea mixture and mix well.
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In another medium bowl, beat egg whites and cream of tartar until foamy. Gradually add remaining 1⁄2 cup sugar in a slow, thin stream, beating until whites form peaks that are stiff but not dry.
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Fold beaten egg whites into chickpea purée. Pour batter into the prepared pans and bake 30 to 35 minutes, or until a toothpick inserted into center of cake comes out clean. Cool 10 minutes on a rack, then turn cake out of pans and onto racks. Cool completely.
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Dribble one tablespoon lemon juice over each round, and sprinkle tops with powdered sugar.