Preheat oven to 350 degrees F. Line six 6-inch cake pans with parchment paper.
Beat egg whites with an electric mixer on medium-high speed until foamy, about 2 minutes. Increase mixer speed to high and continue beating until soft peaks form, about 3 minutes. Gradually add 1/2 cup of the sugar, continuing to beat until stiff peaks form, about 3 minutes more.
In another large mixing bowl, cream butter and the remaining 1/2 cup of sugar on medium speed until light and fluffy and increased in volume, about 5 minutes. Add applesauce, vanilla, and eggs and mix on medium speed for 1 minute more.
Sift together flour, baking powder, and salt. Add to butter mixture and mix on medium speed for 1 minute. Pour in 3/4 cup water and mix to combine. Fold in egg whites with a large spatula.
Divide batter evenly among 6 small bowls. Starting with the lightest color, tint batter with food coloring, whisking in a few drops at a time until the desired intensity is reached. Pour each of the colored batters into a different parchment-lined cake pan (each should be half full) and bake until a toothpick inserted in the center comes out clean, about 12 to 15 minutes.
Stack layers according to the colors of the rainbow, spreading a thin layer buttercream between each. Using an offset spatula, frost the whole cake with a thin layer of buttercream.
Place cake on a freezer-safe plate. Freeze for 15 minutes, or until buttercream is firm. Remove from freezer and cover cake in sprinkles. See how to cover a cake in sprinkles.