Preheat oven to 350°.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour and next 5 ingredients (flour through salt); make a well in center of mixture. Combine apple butter and next 4 ingredients (apple butter through egg whites) in a large bowl; stir well with a whisk. Add apple butter mixture to flour mixture, stirring just until moist. Fold in carrot.
Pour batter into 2 (8-inch) round cake pans coated with cooking spray. Bake at 350° for 35 minutes or until a wooden pick inserted in center comes out clean. Cool in pans 10 minutes on a wire rack; remove from pans. Cool completely on wire rack.
Place 1 cake layer on a plate; spread with 2/3 cup frosting, and top with remaining cake layer. Spread the remaining frosting over top and sides of cake. Store cake loosely covered in refrigerator.