Preheat oven to 400 F and line baking sheet pans with parchment.
Bring butter, sugar, salt, and water to a full boil in a medium saucepot. Remove pot from heat and using a wooden spoon, quickly beat in flour until mixture pulls away from sides of the pot.
In a mixing bowl, whisk eggs and egg white together until well blended.
With wooden spoon, beat eggs, about 1/4 cup at a time, into the flour mixture in the pot, continuing to beat until a ribbon-like batter forms after each addition.
Continue to beat in egg until batter seems as if it has absorbed enough and the dough is still stiff enough to pipe rounds from a pastry bag that will hold their shape.
Fit a piping bag with a 1/2-inch round or fluted tip of choice and fill halfway full of choux paste. Pipe rounds of desire size 1-inch apart on the prepared baking sheet pans.
Bake for 25 minutes then test. Shells are done when they are golden brown and dry in the middle. Shells can be moved to a warming oven at 180 F., if necessary, to allow the centers to dry.