Preheat oven to 350°F. Grease one 8-inch round cake tin and line the bottom with a circle of parchment paper. Dust with flour, tapping out any excess.
In a medium mixing bowl, whisk cake flour, all-purpose flour, sugar, baking powder, baking soda, and salt.
In the bowl of a stand mixer with fitted with a whisk attachment, add one cube of butter to the dry ingredients and beat on medium speed. Repeat until all the butter is added, the mixture looks sandy, and there are no lumps.
In a liquid measuring cup, whisk egg, egg white, buttermilk, oil, and vanilla extract.
Switch the mixer to the beater attachment and, mixing on low speed, slowly pour the wet ingredients into the dry ingredients. Scrape the bottom and sides of the bowl and keep beating on medium for another 15 seconds to incorporate the entire mixture. Mix in sprinkles until evenly distributed.
Fill the prepared tin with the batter. Bake on the middle rack for about 38-40 minutes, or until the top is golden brown and springs back when lightly pressed. A wooden skewer should come out clean (or with just a few moist crumbs) and the internal temperature should read about 195-200°F on a digital instant-read thermometer.
Transfer the tin to a wire rack and allow to cool for 5 minutes before loosening the cake from the sides of the tin with a mini offset spatula or a butter knife and turn out onto the wire rack to cool completely.
Once the cake has cooled, make the frosting. In the bowl of a stand mixer fitted with the beater attachment or in a medium mixing bowl with a hand mixer, beat the butter and powdered sugar until lightened in color and fluffy. Add in vanilla and cream cheese and beat on low just until combined.
Frost the cake and top with rainbow sprinkles and enjoy!