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1
Position a rack in the center of the oven.
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2
Line a large baking sheet with parchment paper or a silicone baking mat.
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3
Bring the milk and butter to a simmer in a heavy-bottomed medium saucepan over medium heat, stirring constantly to be sure the butter is completely melted by the time the milk simmers.
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4
Add the flour, all at once, and stir with a wooden spoon to make a thick paste.
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5
Decrease heat to low.
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6
Stir constantly until the paste comes together into a ball and films the bottom of the saucepan, about 1 minute.
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7
Adjust the heat as necessary so the paste cooks without burning.
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8
The idea here is to force off excess moisture, in the form of steam, from the paste in order to make a crisper pastry; remove the saucepan from the heat.
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9
Whisk 4 of the eggs in a bowl to combine them.
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10
One-fourth at a time, stir the beaten eggs into the hot dough in the saucepan, and stir well until the dough comes together into a glossy mass.
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11
Stir in the Gruyere, rosemary, mustard, salt, and pepper.
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12
Transfer the warm dough to a pastry bag fitted with a 1/2-inch plain tip.
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13
Pipe 25 walnut-size balls of dough, about 1 inch apart, onto the baking sheet.
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14
Or drop the dough from a teaspoon onto the sheet.
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15
Beat the remaining egg well with a pinch of salt.
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16
Lightly brush some of the egg glaze on the mounds of dough, being sure that the egg does no drip down onto the sheet.
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17
Bake until the balls are puffed and golden brown, 20-25 minutes.
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18
If the gougeres have not baked long enough, they will deflate when taken from the oven, so bake for at least 20 minutes before checking them.
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19
Remove the sheet from the oven.
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20
Pierce each gougere with the tip of a small knife (this releases the steam from the interiors of the puffs and helps crisp them).
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21
Return to the oven and continue baking until the gougeres are crisp, 5-8 minutes.
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22
Let cool briefly on the baking sheet.
