Preheat the oven to 375°. Line a 9-inch pie plate with pie dough and stick it in the freezer whiel you prepare the other quiche ingredients.
In a large skillet, heat olive oil over medium heat. Add the asparagus spears and spinach. Cook until asparagus spears are slightly tender and spinach is wilted. Transfer spinach to a collader. Press firmly with the back of a spoon to squeeze out as much liquid as possible. Set aside.
In a large bowl, whisk together eggs and milk. Stir in the feta and mozzarella cheese. Season with salt and pepper to taste.
Remove pie crust from freezer. Place asparagus pieces, spinach and green onions on the bottom of the crust. Pour the egg and cheese mixture over the vegetables and into the crust.
Back the quiche for 45 minutes or until quiche is set and slightly golden brown. Let quiche stand for 15 minutes before serving.