Step 1 Heat the oven to 425 degrees. In a large, heavy-bottom saucepan, bring the water, milk, butter, salt and sugar to a boil. Quickly stir in the flour, making sure to remove any lumps. Reduce the heat to low and continue to stir until the mixture forms a thin film on the bottom of the pan, 2 to 3 minutes. Remove from heat.
Step 2 Transfer the mixture to the bowl of a stand mixer and beat with the paddle attachment on medium speed until the mixture cools. With the mixer running, beat in the 5 whole eggs, one at a time, until fully incorporated.
Step 3Transfer the dough to a large piping bag fitted with a medium (one-half inch) round tip. The batter will make about 1 1/2 dozen large (2 inches wide by 1 inch high) round buns for the base and 1 1/2 dozen small (1 inch wide by 1 inch high) round buns for the top. Pipe the buns out on parchment-lined baking sheets. You will need two baking sheets, one each for the large and small buns. Space the buns about 2 inches apart for the larger buns and 1 1/2 inches apart for the smaller buns.
Step 4Thin the remaining egg white with a tablespoon of water to make an egg wash and very gently brush the wash over the buns. Bake one sheet at a time until the buns are puffed and lightly golden, about 10 minutes for the smaller buns and slightly longer for the larger. Reduce the heat to 350 degrees and continue to bake until the buns have dried out and the insides are set, about 10 to 15 minutes. (Increase the heat back to 425 degrees and bake the second sheet in the same manner.) Cool the buns and set aside.