Place half of the bread cubes in the casserole dish.
6 cups cubed brioche bread
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.
2 large eggs, 1 cup brown sugar, ¼ cup granulated sugar, 1 tablespoon ground cinnamon
Pour filling evenly over bread cubes.
Top filling with other half bread cubes.
For the egg mixture, whisk together 2 egg yolks, eggs, half and half, and vanilla extract in the same bowl you used for the filling.
2 large eggs, ½ teaspoon vanilla extract, 2 ½ cups half-and-half
Pour egg mixture over bread cubes, and gently press bread cubes into mixture. Let the bread pudding sit for 10-15 minutes while you preheat the oven.
Preheat oven to 325 F.
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.