Stir together the water, butter, sugar, and salt in a saucepan. Bring to a boil over medium-high heat.
Remove the pan from the heat, and add the flour all at once. Working quickly, stir vigorously until thoroughly combined.
Return the pan to medium heat, and continue stirring until the mixture is smooth and forms a thin crust on the bottom of the pan. The dough will be thick. (This should only take 2-3 minutes.)
Remove the pan from the heat, and allow the mixture to cool for about 5 minutes. Then transfer the mixture to a mixing bowl and let it continue cooling until it is still warm to the touch. This will take 5-10 minutes. Aim for a temperature below 125°F (51°C).
Using an electric mixer on medium speed, add the eggs one at a time, mixing well after each addition. (About 1 minute each) The pastry should be soft but able to be piped.
Transfer the batter to a pastry bag fitted with a large star tip, filling it about halfway. Pipe 3-inch (7.5cm) circles of dough onto the parchment squares. Allow the piped dough to rest while you heat the oil.
Fill a deep pan with 2 to 3 inches of oil. Heat to 375°F (190°C) over medium heat.
Use tongs to grab a corner of a parchment square and gently place into the oil with the parchment side facing up. Repeat with another 1 or 2 crullers, being sure not to crowd them in the pan.
After about a minute, use tongs to remove the parchment paper. Cook about another 2-3 minutes, and then use tongs to turn the crullers over and cook 2-3 more minutes until lightly browned. Keep an eye on the oil temperature and adjust the heat as needed.
Remove the crullers from the oil using tongs or a large slotted spoon. Place them onto a wire rack placed over a baking sheet.
Repeat with the remaining dough.
To make the glaze:
Combine the glaze ingredients, using enough milk to make a thin, dip-able consistency.
Dip the warm crullers into the glaze, and then return them to the rack to allow the glaze to set.
Notes
Crullers are best enjoyed fresh but can be stored in an airtight container up to 2 days.