For the cake:
2 cups (8 ounces) cake flour
1¾ cups plus 1½ teaspoons (8 ounces) all-purpose flour
2¼ teaspoons baking powder
1 cup (2 sticks) unsalted butter, plus additional for pans
3 cups (21 ounces) granulated sugar
¾ teaspoon salt
2 teaspoons vanilla bean paste
1 teaspoon pure almond extract
1 cup (8 ounces; about 7 eggs) egg whites
1½ cups (12 ounces) whole milk
Red, orange, yellow, green, blue and purple gel food coloring
For the frosting:
3¼ cups (6 sticks plus 4 tablespoons) unsalted butter, at room temperature
6 cups confectioners’ sugar, sifted
Pinch of salt
2 vanilla beans, split lengthwise
2 teaspoons vanilla extract
¼ cup heavy cream
Heat oven to 350º. Brush the bottoms and sides of 8-inch cake pans with melted butter and line with parchment paper. (I have three cake pans and simply had to repeat this process after the first layers had baked bad cooled.) In a large bowl, sift together the cake flour, all-purpose flour and baking powder. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, combine the butter and sugar and beat on medium speed until light and fluffy. Mix in the salt, vanilla bean paste and almond extract. Reduce speed to low and gradually add the egg whites, scraping often, and mix until incorporated. Alternately add the flour mixture and milk to the mixer in two batches, starting with the flour. Scrape down the bowl after each addition and beat until thoroughly combined. Increase speed to medium-high and beat for about 20 seconds, then stop and scrape down the sides of the bowl.
Divide the batter evenly into six bowls (if you have a scale, you’ll want about 11 ounces in each bowl). Tint the batter until your desired colors are reached. Remember, you can always add more but can’t take it away. Once the batter is tinted, pour into prepared cake pans and bake for about 15 minutes. Place on a wire rack and cool for about 15 minutes before removing from the pan, then cool completely on a wire rack. If not frosting right away, the cakes can be tightly wrapped (once completely cool) in plastic wrap and stored at room temperature overnight or frozen for up to 5 days.
To make the frosting, add the butter to the bowl of an electric mixer fitted with the whisk attachment. Whip on medium-high speed until smooth, about 1 minute. Add the confectioners’ sugar to the bowl and mix on medium-low speed just until incorporated. Scrape the seeds from the vanilla beans into the bowl and add in the salt. Continue beating on medium-high speed until smooth, 1-2 minutes. Add the vanilla extract and heavy cream and beat on low speed just until incorporated. Increase the mixer speed to high and beat until light and fluffy, scraping down the bowl as needed, about 4 minutes.
To assemble the cake, place the bottom layer onto a cake board. Spread a scant cup of frosting almost to the edges and top with the next layer. Repeat until all layers are stacked. Frost the top and sides as desired.