Preheat the oven to 350°F and butter the sides of two 9-inch cake pans. Cut two 9-inch circles of parchment paper and line the bottom of each pan with it.
Put the 4 large eggs and the egg white (room temperature) into the bowl of a stand mixer and with the whisk attachment, mix them on medium to high speed for about 1 minute.
Gradually add the granulated sugar to the eggs while it is mixing. Only add 1-2 tablespoons at a time over about a 2-minute time frame.
Let the eggs and sugar mix for about 3-4 minutes until it has doubled or tripled in volume, and it becomes very white.
While the eggs are mixing, melt the butter and heat up the buttermilk until it is hot.
Sift the flour, salt, baking soda and baking powder together and add half of it to the eggs and sugar batter after the eggs and sugar have mixed. Mix it on low speed until combined then add the other half, mixing it until it is just combined.
While the mixer is on, slowly pour the melted butter and hot buttermilk into the cake mixture then add the oil and vanilla extract. Don’t overmix.
Pour the batter into the prepared pans and sprinkle the top of each cake evenly with ¼ cup of sprinkles on each cake. Take a knife and give it a swirl.
Bake the cakes in the oven for 30 minutes or until golden brown and a toothpick inserted in the center comes out clean.
Remove the cakes from the oven and let them cool for 15 minutes.
Loosen the cakes from the sides of the pan then carefully invert them onto a wire cooling rack and remove the parchment paper.
After they have cooled completely, place the first cake layer on the cake platter and frost with your favorite buttercream frosting. Carefully layer the second cake on top and frost it with buttercream also. Generously put multi-colored sprinkles all over the cake, or just on the sides. Serve and enjoy!