Preheat oven to 375 degrees. Place a silpat (a French nonstick baking mat) on a baking sheet. Set aside. In a small saucepan, bring 3/4 cup water, the butter, and salt to a boil. Immediately remove from 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, 30 to 60 seconds. Beat the mixture over low heat for 30 to 60 seconds more to dry it. Make an egg glaze: Whisk 1 of the eggs in a bowl; set aside. With a wooden spoon, beat the remaining eggs into the dough 1 at a time, beating thoroughly after each addition. Beat enough of the reserved beaten egg into the dough until it is shiny and just falls from the spoon, using as much egg as necessary. Beat in the Gruyere cheese. Transfer the dough to a pastry bag fitted with a 1/2-inch plain tip, and pipe 12 mounds about 2 inches in diameter onto the prepared baking sheet. Brush with the egg glaze. Bake the gougeres until puffed and golden, 25 to 30 minutes. Take 1 gougere from the baking sheet, and let it cool for 1 minute. If it remains crisp on the outside, it is done. If not, replace it, and continue baking 5 to 10 minutes longer. Let gougeres cool slightly before removing from baking sheet. Serve warm.;
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