Make the Crullers: Combine the water, butter, sugar, and salt in a large saucepan and bring to a rolling boil over medium-high heat. Immediately remove from the heat, add all the flour at once, and stir hard with a wooden spoon until all the flour is incorporated, about 30 to 60 seconds. Return the pan to the heat and cook, stirring, to evaporate some of the moisture, about 2 minutes.

Scrape the mixture into a standing mixer fitted with a paddle attachment (or use a hand mixer or mix by hand), and mix at medium speed. With the mixer running, and adding 1 egg at a time, add 3 of the eggs, stopping after each addition to scrape down the sides of the bowl. Mix until the dough is smooth and glossy and the eggs are completely incorporated. The dough should be thick, but should fall slowly and steadily from the beaters when you lift them out of the bowl. If the dough is still clinging to the beaters, add another egg and mix until completely incorporated.

Line a sheet pan with parchment paper. Using a pastry bag fitted with a star tip (use a large size, like #12), pipe the dough onto the sheet pan in rows of 2 1/2-inch rings. Freeze them for 30 minutes to make them easier to pick up.

Meanwhile, make the Glaze: Stir together the powdered sugar and milk in a small bowl.

Pour the vegetable oil into a large pot to a depth of 2 inches and heat to 325 degrees F. Working in batches, lift the dough circles off the sheet pan and carefully slip them into the oil. Fry, turning once, until lightly browned. Drain the crullers on a brown paper bag; then dip them completely in the glaze. Let the crullers cool and set before serving.