Prepare oven and baking pan: Preheat oven to 350°F. Butter and flour a 9- x- 13-inch metal baking pan. Set aside.

Combine dry ingredients: Using the paddle attachment, mix the flour, granulated sugar, brown sugar, baking powder, and salt in a stand mixer on low.

Add butter and shortening: Cut the butter and shortening into small pieces, and add to the dry ingredients in the mixer. Turn the mixer on low and mix until combined.

Make egg mixture: Meanwhile, whisk together the buttermilk, oil, eggs, egg yolk, and clear vanilla extract in a separate small bowl.

Make cake batter: Add half of the liquid ingredients to the dry ingredients in the mixer, and combine on low for 2 minutes. Scrape down the sides of the bowl, add half of the remaining liquid ingredients, and combine on low for another 2 minutes. Scrape down the bowl again and add the final amount of liquid ingredients, combining it on low for another 2 minutes. Add ¼ cup of the rainbow sprinkles and fold into batter with a spatula.

Add batter to pan: Pour Cake batter into prepared baking pan. Sprinkle surface with remaining 2 Tbsp. rainbow sprinkles.

Bake cake: Bake, rotating halfway through baking, until a toothpick inserted in the center of the cake comes out dry, about 30 minutes. Allow the cake to cool completely, about 1 ½ hours.

Make the Frosting: Beat together butter and cream cheese on medium in a stand mixer for 3 minutes. Add powdered sugar, and combine on low until just incorporated, then increase the speed to high and beat for another minute. Add the remaining ingredients and beat on medium-high for 5 minutes until light and fluffy.

Decorate cooled cake: Cover the surface of the cake with the Frosting and decorate the surface with more sprinkles.