Preheat the oven to 325°F (163°C) and prepare six 6-inch cake pans by spraying the sides with baking spray and fitting a parchment paper circle to the bottom of each pan.
In a medium bowl, add the flour, baking powder, baking soda and salt. Whisk together and set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter at high speed until creamy, about 2 minutes. Add the sugar and cream it together with the butter at medium-high speed until light and fluffy, about 2 minutes, scraping the bowl and paddle at the halfway point. Turn the mixer to low and add the egg whites one at a time, mixing until they are just combined and scraping down the bowl and paddle as needed. Add the vegetable oil and vanilla, increase the mixer speed to high and beat for 1 minute.
Turn the mixer off and add the flour mixture all at once. Mix the ingredients on low speed until just combined, then slowly pour in the buttermilk. Continue mixing at low speed for about 30 seconds, until the batter is uniform and combined. The batter will be slightly thin.
Divide the batter evenly between six bowls. You should have about 200g of batter per bowl. In the first bowl, whisk in a few drops of red food coloring until you create red colored batter (keep in mind it will darken in the baking process!). Repeat the process with the remaining bowls using orange, yellow, green, blue, and purple food coloring.
Pour each bowl of batter into a separate cake pan and bake for 18-22 minutes, until the layers spring back to the touch and a toothpick inserted comes out clean. Let the cake layers cool in their pans for 5 minutes, then remove them from the pans and allow them to cool completely on a wire rack.