Place the water, butter (cut into pieces) and salt in a heavy saucepan. Bring to a boil. When the butter is completely melted, remove from the heat and add the flour all at once. Mix rapidly with a wooden spatula.
Place the mixture on top of a low flame and "dry" for 1 or 2 minutes, mixing with the wooden spatula. The dough should be soft and should not stick to your fingers when pinched. (This mixture is called the "panade.")
Transfer the panade to a clean bowl. You will notice that the bottom of the pan is covered with a thin crust, an indication that the dough has been sufficiently dried. The eggs are mixed into the panade in the bowl because if they were added in the pan, the white crust at the bottom would break into dried little pieces that would stick in the dough.
Let the dough cool for at least 5 minutes. Add the eggs one at a time, beating carefully after each additions so that the mixture is smooth before the next egg is added.
The dough should be smooth, shiny, and as thick and as heavy as mayonnaise.
To make puffs, fill a pastry bag with the dough and coat a large cookie sheet with butter and flour. Squeeze out puffs about the size of a golf ball or elongated clairs. They can also be formed by dropping spoonfuls of dough on the cookie sheet.
Brush the tops with and egg wash (1 whole egg, beaten), pushing down the "tails."
Drag the tines of a fork to make a design on top of the clairs. Let dry for at least 20 minutes before cooking. (The egg wash gives a shiny glaze, providing it is allowed to dry for a while before baking.)
Bake in a 375 F preheated oven for 35 minutes, or until well puffed and gold. Shut off the heat, open the oven door halfway (to get rid of any steam) and let the puffs cool slowly and dry for 30 minutes. The puffs/clairs will soften and collapse if cooled to fast. Cut into halves to fill or, if you want to, keep them whole.