Preheat the oven to 350 F. Prepare the box mix according to the package directions. Divide the batter between five or six small bowls. Add a small amount of food coloring to each bowl to make different pastel shades. Scrape the colors into small cake pans, pie tins, or other small oven-safe containers. Bake until a toothpick inserted into the center comes out clean and the top springs back when lightly pressed. The baking time will vary depending on how much batter you use and the size of the pan, so watch the cakes carefully and start checking for doneness after 10-15 minutes of baking.
After the cakes are baked, let them cool completely. Crumble the cooled cakes into separate small bowls, then add a spoonful or two of frosting to each bowl. Work the frosting and cake together with your fingertips until the cake is moistened and holds its shape when rolled into a ball. (If you have ever made cake balls before, you are looking for the same texture as cake balls.) If necessary, add more frosting until you get a consistency that's easy to work with, but stop adding frosting before the cake gets greasy or gummy. Roll the cake into small balls of different sizes, and place them on a wax paper-covered baking sheet. Cover the sheet with cling wrap and keep it in the refrigerator until you're ready to make your cake layers. The polka dots can be made up to a week in advance, and kept in the refrigerator or freezer until ready to use.
To Make the Cakes:
Preheat the oven to 350 F. Line three 9x2-inch cake pans with parchment paper, and spray them with nonstick cooking spray. Combine the yolks, 1/2 cup of milk, and lemon extract in a small bowl, and whisk gently to combine. Set aside for now.
Combine the lemon zest and sugar in the bowl of a large stand mixer, and rub them together with your fingers until the sugar is moist and fragrant. Add the flour, baking powder, and salt, and mix together with a paddle attachment on low speed for 30 seconds.
Add the room temperature butter and the remaining 1 cup of milk to the flour, and mix on low speed for 30 seconds, until the dry ingredients are moistened. Turn the speed up to medium and mix for 90 seconds. With the mixer running on medium-low, add the egg yolks in three batches, mixing for 30 seconds after each addition. At the end, stop the mixer and scrape down the bottom and sides with a spatula so that the batter is well-mixed.
Divide the batter evenly between the three cake pans. Take the cake balls and press them into the batter, pressing some all the way to the bottom, some into the middle, and leaving some on top. You don't have to use all of the cake balls, but do use a lot of them—it takes a surprising amount of polka dots to make the cake slices look well-dotted!
Bake the cakes at 350 F for about 25 minutes, until they're puffed and golden on top, and a toothpick inserted into the center comes out with just a few moist crumbs attached. Cool the cakes completely.
To Make the Coconut Buttercream:
Combine the egg whites and the granulated sugar in the bowl of a large stand mixer, and whisk them together. Choose a small saucepan that lets you fit the base of the stand mixer snugly into the top of the saucepan—this is your makeshift hot water bath. (Alternately, you can use a different bowl or an actual bain marie and then transfer the mixture to a mixing bowl once it’s heated.) Add an inch of water to the bottom of the saucepan, and bring the water to a simmer.
Place the mixing bowl on top of the saucepan, making sure that the bottom isn’t in contact with the water, and heat the egg white mixture. Whisk frequently so that the egg whites don’t cook. Continue to heat the whites until they are hot to the touch, and when you rub a bit between your fingers, you don’t feel any grittiness from the sugar. Once the whites are hot, transfer the mixing bowl to your mixer and fit it with a whisk attachment.
Beat the whites on medium-high speed until they are no longer warm to the touch—feel the outside of the bowl, and make sure that it is around room temperature. Depending on your mixer and the temperature of your environment, this may take 15-20 minutes, or more. Reduce the speed to medium-low and add the softened but cool butter in small chunks, a tablespoon at a time, making sure to wait in between additions. It may separate or look a little gloopy at this point—fear not. Once all of the butter is added, increase the speed again and whip until it comes together and is light and fluffy. If, after 5 minutes, it hasn’t come together, refrigerate the mixing bowl for 5-7 minutes, to cool the mixture down, and whip it again.
Add the salt and 2 tsp coconut extract, and mix until well-blended. Taste the frosting and add more coconut flavoring if desired. The buttercream can be made in advance and kept at room temperature if you’re going to use it the same day, or refrigerated. If it’s been chilled, let it sit at room temperature until it softens, then re-whip it to get the fluffy texture back before you use it.
Trim the tops of the cakes so that they are level. Place one cake layer on a cake cardboard (ideally) or a plate. Spread a generous cup of frosting on top of the cake, extending out to the sides. Place a second cake layer on top of the first, and top that one with frosting as well. Finish with the third cake layer, then cover the sides and top of the cake with a thin layer of frosting, then refrigerate until firm.
Once firm, spread the top and sides with the remaining frosting. To get a very smooth finish, I like to run very hot water over an offset metal spatula, wipe it dry, then slowly run it over the sides and top, rinsing frequently. The heat of the metal helps to smooth out the frosting. Finish the cake by pressing Sixlets, sugar pearls, or other sprinkles into the tops and sides of the cake. For the best taste and texture, serve this cake at room temperature.