Preheat oven to 375 degrees F. Grease a 9-inch pie pan. Put the milk, butter, salt, pepper, and nutmeg in a large, heavy saucepan. Cook over medium heat, stirring frequently, until the butter has melted and the mixture comes to a full boil.
Take the saucepan off the heat and add in the flour all at once, stirring hard with a wooden spoon. Return the pan to the heat. Stir vigorously and constantly for 2 to 3 minutes, until the mixture comes away from the sides of the pan. This recipe will give you good arm muscles. (There are recipes for making gougere in a food processor, but that involves a lot of sticky cleanup. A saucepan and spoon are actually easier in the long run.)
Take the pan off the heat again (this time, turn off the burner). Beat the eggs, 1 at a time, into the hot dough, stirring vigorously after each addition until the egg is completely incorporated and the dough is smooth. The dough will be very stiff and thick. Beat in the Parmesan cheese. Fold in chopped ham.
Scrape/pour the dough into the prepared pie pan. Brush with the egg-water mixture and sprinkle with the extra Parmesan cheese, if desired. Bake for 50 minutes without once opening the oven door. The gougere should be quite brown and puffy. Without taking it out of the oven, make a few slashes in the top with a sharp knife; this will let out some of the steam in the center and help the gougere stay puffy once it's out of the oven. Bake for 10 more minutes. Serve the gougere hot, at room temperature, or cold.