Preheat the oven to 375°–400° degrees.
To make an egg wash, beat 1 egg with a pinch of salt and set aside.
For the gougères, combine the water, butter, and ¼ teaspoon salt in a saucepan and bring to a boil. Reduce the heat, add the flour, and whisk until the batter comes away from the sides of the pan. Cook for 1–2 minutes longer.
Transfer to a mixing bowl and beat in three of the eggs, one at a time, beating until smooth after each addition. Add the last egg only if the batter seems too thick. The dough should be like thick cookie dough. Stir in the Gruyère, saving a little to sprinkle on the top before baking.
To finish the gougères, line a baking sheet with parchment paper and pipe or spoon about a teaspoon of batter for each gougère onto it. Brush with egg wash and sprinkle the tops with the remaining cheese. Bake until golden and puffed, about 12–16 minutes. Serve immediately.