For the Cake: Set oven rack in middle position. (If oven is too small to cook both layers on a single rack, set racks in upper-middle and lower-middle positions.) Heat oven to 350 degrees. Spray two 9-inch round cake pans with nonstick cooking spray; line the bottoms with parchment or waxed paper rounds. Spray the paper rounds, dust the pans with flour, and invert pans and rap sharply to remove excess flour.
Mix cake flour, sugar, baking powder, and salt in bowl of electric mixer at slow speed. Add butter and vegetable shortening; continue beating at slow speed until mixture resembles moist crumbs, with no powdery streaks remaining.
Add ½ cup of milk to crumbs and beat at medium speed (or high speed if using handheld mixer) for 1½ minutes. Add ½ cup of water and beat 30 seconds more. Stop mixer and scrape sides of bowl. Return mixer to medium (or high) speed and beat 20 seconds longer.
Now set this mixture aside.Clean the paddle wheel thoroughly.Beat the egg whites into soft mound of white peaks.
Gently fold the beaten egg whites into the batter.Do the folding with a spatula.
Divide batter evenly between two prepared cake pans; using rubber spatula, spread batter to pan walls and smooth tops. Arrange pans at least 3 inches from the oven walls and 3 inches apart. (If oven is small, place pans on separate racks in staggered fashion to allow for air circulation.) Bake until thin skewer or toothpick inserted in the center comes out clean, 23 to 25 minutes.
Let cakes rest in pans for 3 minutes. Loosen from sides of pans with a knife, if necessary, and invert onto wire racks. Reinvert onto additional wire racks. Let cool completely, about 1½ hours.