Preheat oven to 375 degrees F. Butter a large baking or cookie sheet.
In a saucepan over medium-high heat, combine water, butter, and salt; bring just to a boil and remove from the heat immediately. Add all of the flour at once (flour stirred in all at once will quickly swell the starch and dry the dough), stirring vigorously with a wooden spoon until blended. The flour absorbs the liquid rapidly, and must be mixed thoroughly until there are no lumps left.
When the flour is incorporated, return the pan to the heat for several minutes, stirring constantly until the mixture starts to dry out and pull away from the edge of the pan. Remove from the heat. Let the dough cool until it is no longer hot enough to cook the eggs (if the dough is too hot, it may cook the eggs - poke the dough with your finger to test).
Add the eggs one at a time, mixing each one until well blended. The dough should be soft and shiny. Add the blue cheese. Season with black pepper and more salt if needed.
Using two teaspoons: With one spoon, scoop up about 2 to 3 teaspoons for a cocktail-size puff (walnut size). With other spoon, push batter off first spoon and drop in a mound onto prepared baking sheet.
Using plastic bag: Place dough into bag. Cut the end off one corner with scissors, to make an opening about 1/4 inch. Squeeze the mixture onto the pans.
Using pastry bag: Put a 1/4-inch star tip into your pastry bag . Fill the bag with the dough. Squeeze the dough out in a circle motion to create small silver dollar-sized circles.
Leave about 2 inches between the gouges on the baking sheet, as the mixture will spread during baking. Do not worry if they don't look neat, they tend to round out as they rise in the oven.
Using a pastry brush, paint tops with a glaze of the beaten egg. Also gently flatten anypointed tops with your fingers. NOTE: You may also sprinkle black pepper over the top.
Bake for about 30 minutes, or until puffed and golden brown. To be sure that the gouges are fully baked, pierce the top of one with a small sharp knife. The inside should be hollow and dry. Do not under-bake them, as they may collapse.
Remove the pans from the oven and serve the gouges while hot. The puffs are at their best when served warm from the oven, but are also delicious at room temperature. For best results, use puffs with 24 hours, or wrap airtight and freeze.
NOTE: I prefer to prepare the dough 2 to 3 hours in advance. Make the dough balls and cover. When ready to bake, just pop them into a preheated oven. The dough may be made a day ahead and kept covered in the refrigerator. You may also bake and freeze them. Thaw at room temperature and refresh in a 350 F. oven for about 10 to 15 minutes.
Makes 24 (3-inch) gougeres.
* I find that using bread flour lightens the puffs for higher and lighter puffs.
Blue Cheese Gougeres Recipe: https://whatscookingamerica.net/Appetizers/BlueCheeseGougere.htm