For the vanilla cake: Preheat the oven to 350 degrees F. Prepare two 8-inch cake pans with baking spray and line with parchment rounds.
Combine the eggs, 1/4 cup of the milk, and the vanilla in a small bowl. Whisk by hand until lightly combined.
In a stand mixer on low speed, combine the flour, sugar and baking powder. Add the butter and the remaining 3/4 cup milk, and mix on low speed until combined. Increase the speed to medium and beat for 2 minutes, until light and fluffy. Scrape down the bowl with a rubber spatula. Add the egg mixture in three parts, mixing on medium speed for 30 seconds after each addition. Scrape down the sides of the bowl.
Pour the batter evenly into the two prepared cake pans, scraping the bowl. Spread the batter with an offset spatula so it is evenly in the pan.
Bake until a wooden toothpick inserted in the center of the cake comes out clean and the cake springs back when pressed lightly in the center, 25 to 30 minutes. (The cakes will shrink away slightly from the sides of the pan.)
Let the cakes cool in the pans on a wire rack for 10 to 15 minutes. Run a small offset spatula around the outside of the cakes to loosen, then carefully flip them out of their pans and onto the wire rack. Place the cakes right-side up on the rack and let cool completely before icing.
For the buttercream icing: Make sure to have a completely clean and dry bowl when you start your process. Any fat or liquid at all in the bowl will stunt the protein development of the albumen (egg white protein), and you will not have a proper meringue at the end.
Begin by whipping the egg whites by themselves (no sugar or butter yet) on medium-low speed in a stand mixer until the whites are foamy and opaque¿it should look like the head on glass of beer. Increase to medium speed and slowly start adding the sugar until all the sugar is incorporated.
When all the sugar is incorporated, increase the mixer speed to high and whip until shiny and stiff. You now have a meringue. You know your meringue is done when you pull out the whip, hold it horizontal, and you have what looks a "sparrow's beak" on the end of the whip.
Replace the whip, turn the mixer on medium and start adding the butter, a bit at a time. Once all the butter is incorporated, continue whipping on medium speed until the butter has completely emulsified into the meringue. When you first add the butter, your meringue will break down and look like cottage cheese but will continue to emulsify into a smooth buttercream. Depending on the weather, the buttercream could take anywhere from 5 to 15 minutes to form. When the buttercream is ready, the mixture will be homogeneous and smooth¿and tasty.
Remove the buttercream from the bowl and place it in an airtight container. Buttercream can be kept at room temperature for a few days or in the fridge for a week or two, but always use soft buttercream when icing a cake. To warm up the buttercream, put it back in the mixer using a rubber spatula. Begin remixing the buttercream, using the whip attachment on high speed while applying direct heat to the outside of the bowl with a propane torch that can be found at any hardware store.
Assemble the cake: Start by using an offset spatula to spread a layer of icing onto one cake layer. Set the second cake layer on top of the first. Ice the entire assembled cake.
Don't worry about using raw egg whites in your buttercream, the sugar cooks the egg whites and makes them perfectly safe to eat. If you are still uneasy about this, use pasteurized egg whites.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.