Rainbow Cake

FOR THE CAKE: Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Grease three 9-inch round cake pans, line with parchment paper, grease parchment, and flour pans.

Whisk 1 cup milk, 6 egg whites, and 1 teaspoon vanilla together in 2-cup liquid measuring cup. Using stand mixer fitted with paddle, mix 2¼ cups (9 ounces; 255 grams) flour, 1¾ cups (12¼ ounces; 347 grams) sugar, 4 teaspoons baking powder, and 1 teaspoon salt on low speed until combined. Add 12 tablespoons butter, 1 piece at a time, and mix until only pea-size pieces remain, about 1 minute. Add half of milk mixture, increase speed to medium-high, and beat until light and fluffy, about 1 minute. Reduce speed to medium-low, add remaining milk mixture, and beat until incorporated, about 30 seconds (batter may look slightly curdled). Give batter final stir by hand.

Divide batter evenly among three bowls. Add ½ teaspoon red gel food dye to one bowl, ¼ teaspoon yellow to second bowl, and ¼ teaspoon orange to third bowl. Stir each to combine. Transfer each colored batter to separate prepared pans. Smooth tops with rubber spatula and gently tap pans on counter to release air bubbles. Place 2 pans on upper rack and one pan on lower rack. Bake until toothpick inserted in center comes out clean, 18 to 22 minutes, switching and rotating pans halfway through baking. Let cakes cool in pans on wire rack for 10 minutes. Remove cakes from pans, discarding parchment, and let cool completely on rack, about 2 hours.

Repeat steps 1 through 3 with remaining ingredients, using ½ teaspoon blue, ½ teaspoon green, and ½ teaspoon purple gel food dye in each bowl of batter. (Cakes can be stored at room temperature for up to 24 hours or frozen for up to 1 month; defrost cakes at room temperature.)

FOR THE FROSTING: Using stand mixer fitted with paddle, beat 4 sticks butter, ¼ cup cream, 1 tablespoon vanilla, and ¼ teaspoon salt on medium-high speed until smooth, about 1 minute. Reduce speed to medium-low, slowly add 4 cups sugar, and beat until incorporated and smooth, about 4 minutes.

Increase speed to medium-high and beat until frosting is light and fluffy, about 5 minutes.

Repeat steps 5 and 6 with remaining frosting ingredients. (Frosting can be refrigerated for up to 3 days; let soften at room temperature, about 2 hours, then rewhip in two batches on medium speed until smooth, 2 to 5 minutes.)

Line edges of cake platter with 4 strips of parchment to keep platter clean and place small dab of frosting in center of platter to anchor cake. Place purple cake layer on platter. Spread ¾ cup frosting evenly over top, right to edge of cake. Top with blue cake layer, pressing lightly to adhere, and repeat process with green, yellow, orange, and red layers, pressing lightly to adhere and spreading ¾ cup frosting evenly over each layer. Place ¾ cup frosting in center of top of cake and spread to outer edges, letting any excess hang over edges; smooth top. Spread half of remaining frosting along sides of cake with short side-by-side strokes until entire side is covered with thin coat of frosting. Refrigerate cake until frosting is set, about 15 minutes.

Spread remaining frosting evenly over top and sides of cake. Carefully remove parchment strips before serving. Cake can be refrigerated for up to 24 hours; bring to room temperature before serving.