Preheat the oven to 400 degrees . Grease a 9-inch glass pie dish with 1 teaspoon olive oil. Using a food processor fitted with a shredding attachment, shred the potatoes; transfer to a large bowl and stir in the onion, cornstarch, 1/2 teaspoon salt and 1/4 teaspoon pepper.
In a large skillet, heat 1 tablespoon olive oil over medium heat. Add the potato mixture, flattening it into a 10-inch layer, and cook until the bottom is golden, about 10 minutes. Loosen, then invert onto the pie dish and mold into a crust. Bake about 20 minutes.
Meanwhile, preheat the skillet over high heat. Add the spinach and 1/4 cup water and cook, stirring, until wilted; drain. Squeeze out any excess water. Chop the spinach.
In the same skillet, heat the remaining 1 tablespoon olive oil over medium heat. Add the mushrooms and cook, stirring occasionally, until browned, about 6 minutes. Add the spinach; season with salt. Spoon half of the spinach mixture into the crust, add the cheese, then top with the remaining spinach mixture.
In a bowl, whisk together the eggs, egg whites, milk, 1/4 teaspoon salt and 1/4 teaspoon pepper. Pour into the pie dish and bake until set, 20 to 25 minutes.