Preheat oven to 400 degrees F (200 degrees C).
Fit pie crust into a 9-inch pie dish.
Whisk eggs, milk, parsley, garlic, salt, black pepper, and nutmeg in a bowl.
Gently combine spinach, mushrooms, onion, and feta cheese in a separate bowl. Spread spinach-mushroom mixture in the prepared pie dish; top with half the Swiss cheese.
Pour egg mixture evenly over the filling, swirling egg mixture in bowl to spread seasonings through the eggs; top the quiche with remaining Swiss cheese. Place quiche on a baking sheet.
Bake in preheated oven until the quiche is lightly puffed and browned, 45 to 50 minutes. A toothpick inserted into the center of the filling should come out clean. Cool for 30 minutes before serving.
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