Preheat oven to 350 degrees F. Lightly coat an 8x8x2-inch baking pan with cooking spray. Drizzle bottom of pan with melted butter; sprinkle with brown sugar. Arrange pineapple slices in pan; sprinkle with crystallized ginger.
In a large bowl stir together flour, granulated sugar, apple pie spice, baking powder, baking soda, and salt. Add carrots, oil, and milk, stirring until moistened. In a medium mixing bowl beat egg whites with an electric mixer on medium speed until stiff peaks form (tips stand straight). Fold beaten egg whites into carrot mixture. Pour batter into pan over pineapple slices, spreading evenly.
Bake for 35 to 40 minutes or until a wooden toothpick inserted near the center comes out clean. Cool in pan on a wire rack for 5 minutes. Loosen sides of cake; invert onto a serving platter. Cool for 30 minutes. If desired, sprinkle lightly with powdered sugar. Serve warm.