Put water, butter, sugar and salt in a small pot on low heat. When it starts to boil and the butter has melted add the sifted flour in one go. Stir immediately to create a paste.
Keep cooking the paste while stirring continuously for a few minutes.
Transfer in the bowl of the stand mixer with paddle attachment. Alternatively it can be done manually in a big bowl, using a wooden spoon.
Work at low speed until the steam stops coming out of the bowl.
Weight the eggs and beat them slightly (no need to whisk them). This way it will be easier to add small amounts to the paste.
Slowly start adding the eggs bit by bit. Always wait for the previous batch to be absorbed before adding the next one. The quantity of eggs is indicative. When you are close to finish the quantity of eggs check the consistency of the paste. Dip the wooden spoon in paste, lift it up and roll it so that the past falls down. The paste should fall slowly and create a triangle under the spoon.
This is the key to make a good bignet.
Transfer the choux paste in a pastry bag with big round tip. Line a baking tray with parchment paper (or better, with a silicone baking mat).
Using the pastry bag create the bignet, 3 or 4 cm in length (1 – 1.5 in).
For this amount of paste you will probably need 2 trays, or bake in two batches.
Bake in the preheated oven at 190° C (375° F) for approx 23-25 minutes.
Torn the oven off and leave the bignet to dry in the oven for 30 minutes with the door slightly open.
Let them cool off completely before dipping them in the caramel.