Bring water, butter, sugar and salt to a boil over high heat. Turn off heat and dump in the flour. Use a spoon or a spatula to stir it all together. Bring it back to medium heat and cook about 1 minute, until mixture forms a cohesive ball and leaves a film on the bottom of the pot.
Transfer the dough to a stand mixer fitted with the paddle attachment. Let the dough cool for 5 minutes.
With mixer running on medium speed, add a few tablespoonfuls of the beaten egg and incorporate completely. Repeat with remaining egg, a few tablespoonfuls at a time, until all eggs are incorporated and a smooth batter is formed.
Transfer the batter to a pastry bag fitted with a large (1/2" or larger) star tip. Chill for 15 minutes in fridge.
While dough chills, bring a deep pot with 3 inches of oil to 375ºF. Cut rough 4 x 4" squares of parchment paper.
Pipe chilled dough in 3 1/2" rings on the parchment paper squares.
Use tongs to carefully place crullers parchment-side up in the hot oil. Pull off the parchment paper using the tongs and set aside (you can reuse it with the next batch). Fry about 4 crullers at time but don't overcrowd the pan.
Cook for 3 minutes undisturbed on first side then carefully flip the cruller using tongs and cook 2-3 minutes on second side.
Transfer to wire racks to cool before glazing.