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1
Preheat oven to 350 degrees F. Line six 6-inch cake pans with parchment paper.
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2
Beat egg whites with an electric mixer on medium-high speed until foamy, about 2 minutes.
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3
Increase mixer speed to high and continue beating until soft peaks form, about 3 minutes.
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4
Gradually add 1/2 cup of the sugar, continuing to beat until stiff peaks form, about 3 minutes more.
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5
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.
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6
Add applesauce, vanilla, and eggs and mix on medium speed for 1 minute more.
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7
Sift together flour, baking powder, and salt.
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8
Add to butter mixture and mix on medium speed for 1 minute.
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9
Pour in 3/4 cup water and mix to combine.
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10
Fold in egg whites with a large spatula.
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11
Divide batter evenly among 6 small bowls.
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12
Starting with the lightest color, tint batter with food coloring, whisking in a few drops at a time until the desired intensity is reached.
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13
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.
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14
Stack layers according to the colors of the rainbow, spreading a thin layer buttercream between each.
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15
Using an offset spatula, frost the whole cake with a thin layer of buttercream.
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16
Place cake on a freezer-safe plate.
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17
Freeze for 15 minutes, or until buttercream is firm.
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18
Remove from freezer and cover cake in sprinkles.
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19
See how to cover a cake in sprinkles.