Special equipment: two 9-inch cake pans, one 9-inch pie dish, three 10-inch cake boards, two 6-inch cake boards, 2 piping bags fitted with small round tips, 1 piping bag fitted with a small star tip; a lazy Susan or rotating cake stand
For the cake: Preheat the oven to 350 degrees F. Coat two 9-inch cake pans and one 9-inch pie dish with cooking spray, then line with parchment and spray the parchment.
Combine the cake mix and pudding mix in a stand mixer fitted with the paddle attachment and mix on low speed to blend. With the speed still on low, add the oil, egg whites and 1 1/2 cups water and mix until combined. Raise the speed to medium and mix until increased in volume, lighter in color and smooth, about 5 minutes.
Pour the batter into the prepared pans and smooth the tops with an offset spatula or butter knife. Bake until golden on top and the edges pull away from the pans, about 20 minutes for the pie pan and 25 minutes for the cake pans. Transfer to a rack and let cool completely in the pans, then unmold, remove the parchment and refrigerate until firm, about 30 minutes.
For the Swiss buttercream: Meanwhile, bring a few inches of water to a boil in a saucepan that can hold a heatproof stand mixer bowl above the water without the bottom of the bowl touching it.
Add the sugar, lemon juice, salt and egg whites to the mixer bowl and whisk together by hand. Set the bowl above the boiling water and heat, whisking, until the mixture is warm to the touch and the sugar is dissolved. Transfer to the stand mixer fitted with the whisk attachment and beat at medium-high speed until cool and the whites hold stiff peaks, 10 to 15 minutes.
With the motor running on medium-high speed, add the butter a couple of pieces at a time, making sure the pieces are completely incorporated before adding more. After all the butter is added, continue beating on medium-high speed. The mixture will deflate and appear curdled. Continue beating until the buttercream comes back together to form a smooth and spreadable consistency. Slowly drizzle in the vanilla while beating until smooth and light. (If the frosting is very soft or begins to break, refrigerate until set but still spreadable and then beat until light before using.)
Scoop 2 cups of the buttercream into a medium bowl and mix in 3 drops of deep pink gel food coloring; the buttercream should be the same color as the pink animal crackers. Transfer 1/4 cup of the pink buttercream to a piping bag fitted with a small round piping tip. Cover the remaining pink buttercream in the bowl with plastic wrap.
Scoop 1/2 cup of the white buttercream into a medium bowl and mix in 1 drop royal blue and 2 drops neon brite green gel food coloring to make teal. Transfer to a piping bag with a small star tip. Transfer 1/4 cup white buttercream to a piping bag fitted with a small round tip. Cover the remaining white buttercream in the bowl with plastic wrap.
For the assembly: Mix 1 drop pink food coloring into the melted candy. Line a baking sheet with parchment. Put 8 animal crackers--both pink and white--on the baking sheet flat-side up. Spoon a small amount of melted candy onto each cracker and press on a chocolate-covered biscuit stick so the cracker is glued to the top third of the stick. Repeat with 8 more sticks and crackers, attaching the crackers to the bottom third of the sticks. Refrigerate until set, about 5 minutes.
For the carousel base: If the tops of the cakes have domed, carefully trim them with a long serrated knife so that they are flat on top. Place one of the cake rounds on a 10-inch cake board. Spread the top with 1 cup white buttercream and cover with 3 tablespoons sprinkles. Top with the second cake round. Frost the whole cake with 1 cup white buttercream to crumb coat. Refrigerate until firm, about 30 minutes.
Frost the cake with the reserved pink buttercream. Pipe white lines of buttercream down the sides of the cake every 2 inches to create panels. Press white animal crackers into every other panel. Pipe a shell border around the top and bottom edge of the cake with the teal buttercream and top the border with sprinkles. Refrigerate the cake until ready to assemble.
For the carousel top: Wrap a 10-inch cake board with plastic wrap, making sure the plastic on one side is smooth and taut. Place the pie-shaped cake layer on the white side of a 6-inch cake board with the wider side of the cake facing up. Frost the cake with a thick layer of white buttercream. Put the smooth side of the 10-inch board against the top of the cake and flip it over so the 6-inch board is now on top. Frost the angled sides of the cake with white buttercream. Pipe pink buttercream lines down the angled sides every 2 inches, creating panels. Refrigerate until firm, about 30 minutes.
Place a second 10-inch cake board on top of the 6-inch board and flip the carousel top back over. Gently peel the plastic-covered board off the top. (If you have difficulty doing this, chill the cake 15 minutes more. You can also gently loosen the edge of the buttercream from the plastic using a warm offset spatula). The top of your cake should be flat and smooth.
Re-melt the melted candy. Pipe a shell border around the edge of the pie-shaped cake with the teal buttercream and top with sprinkles. With kitchen shears, trim 4 strawberry cream-covered biscuit sticks to 3 inches, gather them in a bundle and press them into the top of the center of the cake; they should stand upright like a column. Dip the coated ends of 10 strawberry biscuits in the melted candy and insert the uncoated ends around the rim of the cake, angling the sticks toward the center column like the spokes of a wheel. Dip the bottom of the gumdrop in the melted candy and affix to the top of the column. Refrigerate until ready to assemble.
For the center column: Spread 1 teaspoon of white buttercream in the center of a second 6-inch cake board and top with a doughnut. Spread it with buttercream, add another doughnut, more buttercream and the last doughnut. Insert 3 chocolate biscuit sticks through the center holes of the doughnuts and trim to the height of the doughnut stack. Refrigerate until firm, about 30 minutes.
Frost the doughnut stack with white buttercream. Press 8 strawberry biscuit sticks against the side of the doughnut stack 1 inch apart. Press 6 animal crackers into the buttercream between the biscuit sticks, using a little buttercream to get them to adhere if necessary. Refrigerate until firm, about 30 minutes.
To assemble: Carefully transfer the doughnut stack to the middle of the carousel base. Insert the cookie sticks with the animal crackers around the rim of the cake, alternating heights and pushing them 1 inch into the cake. Carefully transfer the top of the carousel to rest on top of the doughnuts (leave the 6-inch board on the underside for support).
Serve the cake on a lazy Susan or rotating cake stand. To serve, remove the carousel top and cut the base into wedges.
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