In a medium saucepan, combine the water, unsalted butter, sugar, and salt. Bring this mixture to a boil over medium heat.
Once boiling, reduce the heat to low. Add the flour all at once, stirring vigorously. Keep stirring until the mixture forms a ball and pulls away from the sides of the pan, which should take about 1 minute.
Remove the pan from the heat and allow your dough to cool for 5 minutes.
Now it’s time to add the eggs. Beat in the eggs and the egg white, one at a time, until the dough is smooth and glossy.
In a deep fryer or large pot, heat the vegetable oil to 370°F (187°C).
Transfer the dough into a piping bag fitted with a large star tip. Pipe rings of dough onto squares of parchment paper.
Carefully lower the dough rings into the hot oil, parchment paper side up. The parchment will detach as they begin to fry; use tongs to remove it. Fry the crullers until they are golden brown, roughly 2 minutes on each side. Then, transfer them to a wire rack to drain.
For the glaze, in a separate bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth and creamy.
Dip the tops of each cruller into the glaze, then set them back on the wire rack to let the glaze set.
These are best served fresh, so enjoy them while they are warm!