Preheat oven to 350°F. Lightly grease three, 8-inch round pans with baking spray or butter and flour, then line the bottoms with parchment paper. (I also used cake strips on my pans for even layers.)
In a large bowl, sift together the flour, baking powder, and salt. Set aside.
In a large mixing bowl with a hand mixer or stand mixer fitted with the paddle attachment, beat the sugar, butter, and oil on medium speed until completely smooth and creamy, about 3 minutes.
Add the egg whites in three parts, mixing on low until fully combined before adding the next. Stop and scrape the bowl down occasionally. Add the vanilla and beat until fluffy, about 20 seconds.
Add one-third of the flour mixture, then mix on low speed until combined. Add half of the buttermilk and mix until combined. Repeat with another third of the flour, the remaining buttermilk, ending with the remaining flour mixture, mixing between each addition. Scrape down the bowl and ensure the batter is well mixed and no lumps remain. With a silicone spatula, fold in the rainbow sprinkles.
Divide the batter evenly among the prepared cake pans and spread into an even layer. Tap each pan several times on a towel on the kitchen countertop to settle the batter and release any big air bubbles. Bake for 32 to 38 minutes, rotating the pans on the oven rack after 20 minutes, until the centers spring back when gently pressed.
Let the cakes cool in the pans for 10 minutes, then turn onto wire cooling racks. Remove the parchment paper.
In a large mixing bowl with a handheld mixer or stand mixer fitted with the paddle attachment, beat the butter and salt on medium speed until very soft and smooth, about 2 minutes.
With the mixer on low speed, gradually add in the confectioners’ sugar, adding the heavy cream 1 tablespoon at a time, as needed, to keep the mixture creamy. Beat in the vanilla. Increase the speed to medium and beat for 1 to 2 minutes, until the texture is smooth and fluffy.
Place one cool cake layer top-side up on a cake stand. Spread 1 cup of frosting over the top of the cake all the way to the edge. Repeat with the remaining two cake layers.
Transfer 1 cup of frosting to a piping bag fitted with a large, open star piping tip (I like the Ateco 869.) Spread the remaining frosting all over the top and sides of the cake.
Press rainbow sprinkles into the bottom of the cake, about 2 to 3 inches up the sides, like a “skirt”.
Pipe dollops in 8 to 10 places around the top edge of the cake. Garnish with more sprinkles if you like.