FOR THE CAKES:
3 cups all-purpose flour
4 teaspoons baking powder
1/2 teaspoon salt
2 sticks (1 cup) unsalted butter, room temperature
2 1/3 cups sugar
5 large egg whites, room temperature
2 teaspoons pure vanilla extract
1 1/2 cups milk, room temperature
Red, orange, yellow, green, blue, and purple gel food coloring
FOR THE FILLING AND CRUMB COAT:
9 large egg whites
1 3/4 cups sugar
2 cups (4 sticks) unsalted butter, cut into small pieces, room temperature
2 teaspoons pure lemon extract
FOR THE FROSTING:
5 large egg whites
1 cup sugar
1 cup (2 sticks) unsalted butter, cut into small pieces, room temperature
1 teaspoon pure lemon extract
DIRECTIONS FOR THE CAKE:
Preheat oven to 350 degrees. Brush six 9-inch-round cake pans (or as many 9-inch cake pans as you have, reusing them as necessary) with shortening. Line bottom of each cake pan with parchment paper; brush again and set aside.
In a large bowl, whisk together flour, baking powder and salt; set aside. In the bowl of an electric mixer fitted with the paddle attachment, cream together butter and sugar. Slowly add egg whites and mix until well combined. Add vanilla and mix until fully incorporated. Add flour mixture and milk in two alternating additions, beginning with the flour and ending with the milk. Mix until well combined.
Divide batter evenly between six medium bowls. Add enough of each color of food coloring to each bowl, whisking, until desired shade is reached. Transfer each color to an individual cake pan. Transfer to oven and bake until a cake tester inserted into the center of each cake comes out clean, about 15 minutes (working in batches if necessary).
Remove cakes from oven and transfer to a wire rack; let cool for 10 minutes. Invert cakes onto a wire rack; re-invert and let cool completely.
Variation: See picture below. Instead of six individual layers, you can bake 2 9-inch round cakes. In the first pan pour about 2/3 of the red and orange batters, then 1/2 of the yellow and green batters, then 1/3 of the blue and purple batters. In the second pan, the order will be reversed (the remaining 2/3 of the purple and blue batters, the remaining 1/2 of the green and yellow batters, the remaining 1/3 of the orange and red batters).
DIRECTIONS FOR THE FILLING, CRUMB COAT AND FROSTING:
Make the filling and crumb coat: Cook egg whites and sugar in a small saucepan over medium heat, whisking constantly until sugar is completely dissolved. Transfer to the bowl of an electric mixer fitted with the whisk attachment; mix on high speed until mixture is room temperature. With the mixer on medium-low, add butter, one piece at a time, mixing well after each addition.
Remove whisk attachment and switch to paddle attachment. Increase speed to high and beat until buttercream comes together, about 5 minutes; beat in lemon extract.
Make the frosting: In a clean saucepan and bowl of an electric mixer, repeat same process as for filling and crumb coat.
ASSEMBLING THE CAKE:
Using a serrated knife, trim tops of cakes to make level. Place four strips of parchment paper around perimeter of a serving plate or lazy Susan. Place the purple layer on the cake plate. Spread buttercream filling over the first layer with a small offset spatula so it extends just beyond edges. Repeat process with blue, green, yellow, and orange layers.
Place the remaining red layer on top, bottom-side up. Gently sweep away any loose crumbs with a pastry brush. Using an offset spatula, cover the top and sides with a thin layer of frosting (also use any of the excess frosting visible between the layers). Refrigerate until set, about 30 minutes.
Using an offset spatula, cover cake again with remaining frosting.
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