For the filling, stir together egg whites from the large eggs (set yolks aside to use in milk mixture), sugars, and cinnamon in a medium bowl until well-combined.
Pour filling evenly over bread cubes.
Top filling with other half bread cubes.
For the egg mixture, whisk together 2 egg yolks, eggs, milk, and vanilla extract in the same bowl you used for the filling.
Pour egg mixture over bread cubes, and gently press bread cubes into mixture.
Bake casserole for about 40-45 minutes, or until lightly browned.
To make the icing, whisk together powdered sugar, milk, and vanilla, adding milk as necessary to make desired consistency.