Sift the flour with a pinch of salt. Put the measured water and butter into a saucepan and bring to a boil.
Remove from the heat and add the flour. Beat well until the mixture is smooth and glossy and leaves the side of the pan clean. Leave it to cool slightly.
Beat the eggs and cheese, if using, into the flour mixture until it's smooth and glossy. Make the gougere by buttering a large shallow ovenproof dish and lightly pressing the choux pastry around the edge, leaving a hole in the middle. Cover and chill while preparing the filling.
Melt the butter in a large saucepan, add the mushrooms, and cook gently for 3 to 4 minutes. Lift out with a slotted spoon and set aside. Add the flour and cook, stirring, for 1 minute. Remove from the heat and gradually blend in the milk. Bring to a boil, stirring until the mixture thickens. Simmer for 2 to 3 minutes.
Return the mushrooms to the pan and add the tarragon, salt and pepper to taste. Pour into the middle of the choux ring and bake in a 425*F. oven for 35-40 minutes, until well risen and golden. Garnish with tarragon, and serve hot. Enjoy!