Step 1Heat the oven to 400 degrees. Line 2 or 3 baking sheets with parchment paper. Bring 1 cup of water to a boil in a medium saucepan with the butter and salt. When the butter is completely melted, remove from the heat, and add the flour all at once. Stir briskly with a wooden spoon until the dough forms a ball, about 1 minute.
Step 2Make a well with the spoon in the middle of the dough, add one of the eggs and beat with the wooden spoon. The consistency will be strange at first -- pieces of the dough will slide off of each other -- but in about half a minute, the consistency will become uniform. Add each egg separately, beating until the consistency is uniform, then add the egg white, and beat until smooth and glossy. Beat in the cheeses.
Step 3Using a teaspoon, drop spoonfuls (about 1 1/2 to 2 inches in diameter) of the mixture onto the baking sheet, leaving about 1 inch between them. In a small bowl, lightly beat together the egg yolk and 1 teaspoon water. Brush the top of each gougere with this mixture. Bake for 10 minutes, then turn down oven to 350 degrees and bake 20 to 25 minutes more, until puffed and golden brown.
Step 4As soon as you remove them from the oven, make a small incision in each puff with a small sharp knife to release the steam (and prevent them from getting soggy inside). Serve immediately, or cool and freeze them, reheating for about 3 minutes in a 350-degree oven.
Thyme: Stir in 1 to 1 1/2 tablespoons chopped fresh thyme as you add the grated cheeses.
Blue cheese: Substitute one-half cup crumbled blue cheese for one-half cup grated Gruyere.
Olive and rosemary: Stir in one-half cup chopped oil-cured pitted black olives and 1 tablespoon finely chopped fresh rosemary as the cheeses are added.
Porcini mushroom: Rehydrate one-half ounce dried porcinis in 1 cup boiling water. Drain, pat dry and finely chop. Stir in with the cheeses.