Preheat the oven to 375°F (190°C). Line your cupcake pan with cupcake liners, and set it aside.
In a medium-sized bowl, whisk together the cake flour, baking powder, baking soda, and salt. Set aside.
In a large mixing bowl or stand mixer, beat the softened butter, sugar, and oil on medium-high speed for about 3-5 minutes, until light and fluffy. The mixture should look pale and creamy, ensuring a soft texture in your cupcakes.
Add the eggs and egg white one at a time, beating well after each addition. Once the eggs are incorporated, add the vanilla bean paste and mix until smooth.
With the mixer on low, add one-third of the dry ingredients to the butter mixture, mixing until just combined. Add half of the buttermilk and mix, then alternate with the remaining dry ingredients and buttermilk, finishing with the last third of the dry mix. Be careful not to overmix, as it could lead to dense cupcakes.
Toss the rainbow sprinkles with 1 tbsp of cake flour, then gently fold them into the batter with a spatula. This helps distribute the sprinkles without them sinking or bleeding color.
Fill each cupcake liner 2/3 of the way full with batter. Bake in the preheated oven at 375°F for 5 minutes, then reduce the temperature to 350°F (177°C) without opening the oven door. Continue baking for an additional 10 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Allow the cupcakes to cool in the pan for 2 minutes before transferring them to a wire rack to cool completely. Once cooled, top with your favorite frosting, like classic vanilla buttercream or my personal favorite whipped vanilla frosting; and add a few extra sprinkles for garnish.