Preheat your oven to 375°F. Place a nonstick skillet on the stovetop.
Grease a 9-inch pie dish / plate or baking dish with cooking spray or butter and set aside.
Add the olive oil to the skillet over medium heat. Once warm, add the diced onion and saute for 2 minutes. Add the sliced zucchini and mushrooms along with a pinch of salt and pepper. Saute for 6-10 minutes, until the zucchini and mushrooms have softened and released most of their water and they start to brown. Add the tomatoes, garlic powder and spinach. Saute for 3-4 minutes so the spinach wilts. Set aside.
In a medium bowl, whisk together the eggs, milk, Greek yogurt, crumbled feta, chopped basil and salt and pepper. Set aside this egg mixture.
Add the cooked vegetables to the greased pie dish and spread it out evenly. Pour the quiche filling (egg mixture) on top of this.
Bake in the preheated oven for 40-55 minutes or until the center of the quiche is firm to the touch and is beginning to turn golden brown around the edges. Remove the dish from the oven and let cool on a wire rack for 10-15 minutes before slicing and serving.