Preheat oven to 350°F. Grease and lightly flour two 8-inch round cake pans.
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.
In a medium bowl, combine flour, 1⁄2 cup sugar, baking powder, and salt. Mix well. Add to puréed chickpea mixture and mix well.
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.
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.
Dribble one tablespoon lemon juice over each round, and sprinkle tops with powdered sugar.