Preheat your oven to 350°F (175°C). This ensures that your cake bakes evenly and rises beautifully.
Grease and flour your cake pans or line them with parchment paper to prevent sticking. This step is crucial for easy release after baking.
In a large mixing bowl, cream together the softened butter and sugar until light and fluffy. This should take about 3-5 minutes. Add in the vanilla extract and mix until combined.
Incorporate the egg whites one at a time, mixing well after each addition. This helps to introduce air into the batter, giving your cake a light texture.
In another bowl, whisk together the flour, baking powder, and salt. Gradually add this mixture to the wet ingredients, alternating with the milk. Begin and end with the flour mixture. Mix until just combined.
Divide the batter evenly into five bowls. Add a few drops of food coloring to each bowl, creating red, orange, yellow, green, and blue batters. Stir gently until the colors are vibrant.
Pour each colored batter into its prepared cake pan. Bake for 25-30 minutes, or until a toothpick inserted in the center comes out clean. Allow the cakes to cool in the pans for 10 minutes before transferring to wire racks to cool completely.
In a large mixing bowl, beat the softened butter until creamy. Gradually add the powdered sugar, mixing on low speed until incorporated. Add the vanilla and 2-3 tablespoons of milk, and beat until fluffy.
Once the cake layers are completely cool, place the blue layer on a serving platter. Spread a generous layer of frosting on top, then add the green layer. Repeat this process, stacking each colored cake layer and frosting in between.
Once all layers are stacked, apply a thin layer of frosting around the sides and on top of the cake to create a crumb coat. Chill the cake for 30 minutes, then add a thicker layer of frosting for a smooth finish.
Decorate the top and sides of the cake with sprinkles of your choice for a festive touch.