Preheat oven to 350 degrees F (175 degrees C). Lightly grease an 8×4 inch loaf pan.
In a large bowl, stir together flour, sugar, baking powder, baking soda, cinnamon, nutmeg and salt. Add milk, applesauce, egg whites, bananas and vanilla extract ; stir just until combined. Pour batter into prepared pan.
Bake in preheated oven for 50 to 55 minutes, until a toothpick inserted into center of loaf comes out clean. Turn out onto wire rack and allow to cool before slicing.