Heat oven to 375° F.
In a large skillet, over medium-low heat, heat the oil. Add the onions, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper. Cover and cook until the onions are softened, 5 to 7 minutes. Add the parsley and cook, covered, for 2 minutes more.
Meanwhile, whisk together the eggs and half-and-half. Stir in the Gruyère, nutmeg, the remaining salt and pepper, and the onion mixture.
Place the piecrust on a foil-lined baking sheet. Scrape the egg mixture into the piecrust; it will be very full. Bake until the filling is set and a knife inserted into the center comes out clean, about 40 minutes. Let rest for 5 minutes. Cut into wedges and serve.
Tip: Tailor this quiche to your liking by stirring in a handful of chopped cooked ham or sauteed vegetables when you add the cheese.