Heat the oven to 375°F/190°C and butter a baking sheet. For the choux pastry, sift the flour onto a piece of paper. In a small saucepan, heat the water, butter, and the salt until the butter is melted, then bring everything to a boil. Immediately remove the pan from the heat, add the flour all at once, and beat vigorously with a wooden spoon until the mixture is smooth and pulls away from the sides of the pan to form a ball. Beat the mixture over low heat for 30 to 60 seconds more to dry it.

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Whisk one of the eggs in a bowl until mixed and set it aside. With a wooden spoon or an electric mixer, beat the remaining eggs into the dough one by one, beating thoroughly after each addition. Beat enough of the fourth reserved egg into the dough so it is shiny and just falls from the spoon; you may not need all of the reserved egg. Beat in the grated Gruyère cheese. Put the dough into the pastry bag and pipe 12 mounds about 2 inches/5cm in diameter on the baking sheet. Brush the puffs with the egg glaze.

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Bake the Gougères until puffed and golden, 25 to 30 minutes. Take one puff from the baking sheet and let it cool for a minute. If it remains crisp on the outside, it is done. If not replace it and continue cooking for 5 to 10 minutes. When done, let the puffs cool slightly, then remove them from the baking sheet. Serve them warm they should be slightly warm in the centre.