Heat the oven to 350 (F).
Combine the butter, shortening, and sugars in the bowl of a stand mixer fitted with the paddle attachment and cream together on medium-high for 2 to 3 minutes. Scrape down the sides of the bowl, add the eggs, and mix on medium-high for 2 to 3 minutes. Scrape down the sides of the bowl once more.
On low speed, stream in the buttermilk, oil, and vanilla. Increase the mixer speed to medium-high and paddle for 4 to 6 minutes, until the mixture is practically white, twice the size of your original fluffy butter-and-sugar mixture, and completely homogenous. Don't rush the process -- there should be no streaks of fat or liquid. Stop the mixer and scrape down the sides of the bowl.
On very low speed, add the cake flour, baking powder, salt and the 1/4 cup rainbow sprinkles. Mix for 45 to 60 seconds, just until your batter comes together. Scrape down the sides of the bowl.
Spray a quarter sheet pan with cooking oil and line it with parchment or a Silpat mat. Using a spatula, spread the cake batter in an even layer in the pan. Sprinkle the remaining 2 tablespoons of rainbow sprinkles evenly on top of the batter.
Bake the cake for 30 to 35 minutes. The cake will rise and puff, doubling in size, but will remain slightly buttery and dense. At 30 minutes, gently poke the edge of the cake with your finger: the cake should bounce back slightly and the cake should no longer be jiggly. Leave the cake in the oven for an extra 3 to 5 minutes if it doesn't pass these tests.
Take the cake out of the oven and cool on a wire rack.