Bring the water, butter, sugar and salt to a simmer. Off heat, add all of the flour and whisk vigorously until the flour is well incorporated. Put the pot back on the heat and stir with a wooden spoon until the mixture breaks away from the sides of the pot and adheres to itself to form a ball-like mass. (This should be very quick and take only a few seconds.)

Remove the pot from the heat and let it cool off a bit. Then add the eggs, one by one, mixing well with a wooden spoon until each is well incorporated into the dough. Let the dough cool completely. It will be quite sticky and soft.

To fry your bignè, take two spoons and scoop up a spoonful of the dough with one of them. Then, passing the dough from one spoon to another, form a roundish little dough. (NB: It will be close to impossible to make perfectly round sphere with this wet dough.) Then flick your dough ball into the fat. Then proceed with the rest of the dough, until your skillet is filled (but not too crowded) with little bignè.

The fat should be only moderately hot at first. The dough balls will puff up almost as soon as they hit the oil. Nudge them gently as they fry. They will rotate very easily. When they have all lost their raw look and are nice and puffy, raise the heat and continue frying over high heat until the bignè are all golden brown. Remove them with a slotted spoon to a platter lined with paper towels or a baking grid to drain and cool. Repeat until you've used up all your dough.

If you have more dough to fry, remove the skillet to a cold burner to let it cool off a bit. (If you add your next batch to oil immediately, the oil will be too hot and the bignè will brown before they have a chance to puff up properly.) After a minute or two, you can add your next batch of bignè into the oil off heat, the oil will still be hot enough to start them cooking. Then put the skillet back on the flame and continue as for the previous batch.

After all your bignè are done, let them cool off completely. At this point, you can simply sprinkle them with powdered sugar and serve. Or, for a richer version, you can fill them first with crema pasticcera (see Notes).