Preheat the oven to 425°F.
In a large bowl, toss the potatoes with oil, 1 teaspoon of the thyme, and rosemary. Arrange the potatoes on the bottom of a large oven-safe skillet or baking dish. Bake the potatoes for 10 to 15 minutes, or until they soften and start to crisp.
Meanwhile, combine the remaining 1 teaspoon of thyme and the mushrooms in a large skillet over medium-high heat. Cook the mushrooms for about 5 minutes, or until the mushrooms are brown and most of their liquid has evaporated. Stir in the vinegar. Add the spinach and cook it for 3 to 4 minutes, stirring constantly, until it is wilted. Remove the mushrooms and spinach from the heat and set them aside.
Remove the crust from the oven and turn it down to 325°F.
In a large bowl, whisk together the eggs and milk. Add the mushroom and spinach mixture into the potato-lined skillet and bake the quiche for 25 minutes, or until the eggs are set and no longer runny in the center.