Coat a 15x10x1-inch or 13x9x2-inch baking pan with cooking spray. Set aside.
In a large mixing bowl, stir together the all-purpose flour, packed brown sugar, granulated sugar, cinnamon, baking powder, and baking soda.
Stir together thawed egg product or egg whites, applesauce, and cooking oil. Stir into flour mixture. Stir in carrot.
Spread the dessert batter into the prepared pan. Bake in a 350 degree F oven until a wooden toothpick inserted near the center comes out clean. (You should allow 25 to 30 minutes for the 15x10x1-inch pan or 35 to 40 minutes for the 13x9x2-inch pan.) Cool the bars completely on a wire rack.
Prepare the Pineapple Frosting. Spread the Pineapple Frosting over the dessert; sprinkle with the chopped walnuts or pecans. Makes 36 bar cookies in the larger baking pan or 12 to 15 snack-cake servings in the smaller baking pan.