Using a stand mixer, beat the egg whites with the whisk attachment until they are stiff and form peaks. This may take a minute or two. Pour the egg whites into another bowl and place them in the refrigerator until you're ready to add them to the batter.
Using the same bowl that you used to beat the egg whites, place the softened butter in and cream the butter for about 2 minutes (using the beater blade attachmenuntil it is white in appearance.
Add the sugar to the butter and beat until fluffy (about another 1-2 minutes).
In a small bowl, combine the flour, tapioca starch (measured carefully*), salt and baking powder. Set aside.
In another bowl, combine the milk and vanilla extract.
Add the flour mixture to the butter/sugar mixture alternately with the milk.
Add the stiffly beaten eggs to the cake batter. Fold the egg whites in gently. Do not overmix at this point. If you do, your cake will become more dense.
Grease and flour 2 9" round cake pans. Pour the cake batter equally into the prepared cake pans.
Bake the cakes at 350 degrees for 25-27 minutes or until the top bounces back when you touch it.
Allow the cakes to cool for 10 minutes, then loosen the edges and remove them from the pans to a wire rack, allowing them to cool completely.
One the cakes are cool, frost with your favorite frosting.
*This is about 6 large egg whites.*Some are having problems with the cake being too dense. This can happen when you mistakenly add too much flour. When you measure the flour, spoon it from the flour container gently into the measuring cup. Do NOT pack it down at all. This technique will help you not put too much flour into the cake.