Preheat oven to 350 F. Spray two 9-inch round cake pans with nonstick cooking spray with flour.
In a large measuring cup, whisk together 1 cup milk, 6 egg whites and 2 teaspoons vanilla. Combine 2 ¼ cups flour, 1 ¾ cups sugar, 4 teaspoons baking powder, and 1 teaspoon salt in the bowl of a stand mixer fitted with the paddle attachment. Beat on low briefly to combine. With the mixer still on low, add 12 Tablespoons butter, a few pieces at a time, and continue beating until the pieces of butter are about the size of peas.
Add all but ½ cup milk mixture to the mixer and beat on medium speed for 1 ½ minutes. The batter should be smooth. Add remaining ½ cup milk mixture and beat for an additional 30 seconds. Scrape down the sides of the bowl, then beat for 20 seconds longer on medium.
Divide the batter between the prepared pans. Use an offset spatula to spread into an even layer. Bake for 22-25 minutes, or until a toothpick inserted in the center comes out clean. Transfer the pans to a wire rack and allow the cakes to cool completely in the pans. Meanwhile, make the blueberry and strawberry syrups.