Preheat oven to 350 degrees F. Butter and flour bottom and sides of a glass 9x13 baking dish.
In a large mixing bowl, combine sifted flour, baking powder, cinnamon, and salt. Whisk together well.
In a stand mixer or separate large bowl with a hand mixer, beat egg whites until firm peak forms (6 to 8 minutes). Gradually beat in sugar. Add egg yolks on at a time beating in between each addition. While mixing, slowly add in applesauce. Beat in vanilla.
Add flour and milk by alternating between the flour mixture and milk. While beating, begin by adding one-third of flour mixture and one-half of the milk ending with the flour mixture. Make sure each addition is mixed well before adding the next addition. Pour batter into pan.
Bake for 20 minutes. Reduce heat to 325 degrees F. Bake for 10 additional minutes. Let cake cool for 15 minutes. Poke holes generously all over the cake with a fork.