Preheat the oven to 350 degrees F. Lightly grease a 9-inch diameter pie dish with oil.
In a large skillet, warm 1 tablespoon olive oil over medium high heat.
Sauté the shallots and mushrooms until the mushrooms start to brown, about 5 minutes.
Stir in the red peppers and garlic. Continue cooking for 1-2 minutes until the mushrooms are caramelized and fragrant.
Stir in the spinach and parsley, giving the greens 1-2 minutes to wilt. Remove the skillet from the heat.
In a large bowl, whisk together the eggs, milk, salt and pepper.
To assemble the quiche, spoon about 3/4 of the mushroom-pepper mixture into the bottom of the pie dish and sprinkle on 1/4 cup feta.
Pour in the egg mixture. Top with the remaining mushroom-pepper mixture and feta.
Bake for 40-45 minutes until just set in the middle. It should be lightly browned, and the edges will pull away slightly from the dish.
Sprinkle with chopped parsley before slicing into pieces and serving.