This bright, multihued cake is a festive addition to any party. For a full-color effect, dye the layers to match a rainbow or let your imagination take flight. Create your own palette by tinting the cake layers to coordinate with the jimmies, nonpareils, or dragees of your choice.
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.