Heat oven to 375° F. Fit the crust into a 9-inch pie plate. Place on a baking sheet.
Heat 1 tablespoon of the oil in a large skillet over medium-low heat. Add the onions and ½ teaspoon each of salt and pepper and cook, stirring occasionally, until soft, 5 to 7 minutes. Stir in the parsley.
In a large bowl, whisk together the eggs, half-and-half, nutmeg, and ¼ teaspoon salt. Stir in the onion mixture and the Gruyère.
Pour the egg mixture into the crust. Bake until a knife inserted in the center comes out clean, 35 to 40 minutes. Let sit for 5 minutes. Drizzle the greens with the remaining tablespoon of oil and sprinkle with ¼ teaspoon each salt and pepper. Serve with the quiche.