Preheat oven to 350 degrees F. Butter two round 8-inch pans. Cut out two rounds of parchment paper the size of the pan bottoms. Line bottoms of pans with parchment, then butter again and dust evenly with flour.

In a liquid measuring cup or medium bowl, whisk together the egg whites, whole egg, ¼ cup of the milk and the vanilla. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment, mix the dry ingredients together on low speed.

Add the butter one piece at a time, about every 10 seconds. Continue to mix on low speed until the mixture is a fine crumbly texture. Add the remaining ¾ cup milk, and mix on low speed to combine. Increase the mixer to medium speed and mix for 2 minutes, scraping the side of the bowl as necessary.

Add half of the egg mixture, beat on medium speed to combine, and then add the remaining egg mixture. Use a rubber spatula to make sure the batter at the bottom of the bowl is incorporated.

Divide the batter between the two cake pans, spreading it evenly.

Bake for 25-30 minutes. Cake is done when a tester inserted into the center has just a few moist crumbs attached.

Let the cake cool in the pans for 15 minutes. Run a knife around the edge of the pans and transfer cakes to wire rack to cool completely before frosting.

In the bowl of a stand mixer fitted with the paddle attachment, whip the butter for 8 minutes on medium speed. The butter will become very pale and creamy.

Add the milk, vanilla bean seeds, vanilla extract, and salt and mix until combined. Add in the confectioners' sugar and mix on low speed for 1 minute, and then on medium speed for 6 minutes. The frosting will be very light, creamy, and fluffy and is best used right away.

Place bottom cake layer on a cake plate and spread 1 cup of frosting on top with a small offset spatula. Gently place 2nd cake layer face down on top. Place a generous scoop of frosting on top, spreading evenly with a small offset spatula and working your way down the sides until you have a thin layer of frosting over the entire cake. Chill until set, about 30 minutes.

Remove from refrigerator and apply a final “coat” of frosting.