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Preheat the oven to 375 degrees F. Prepare a 9 x 13” glass or ceramic baking dish by greasing the bottom and sides with oil or cooking spray.
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Heat the olive oil in a large 12” skillet over medium-high heat. Add onions and garlic, sauté until the onions are tender, about 3 minutes. Add mushrooms, sauté until mushrooms are tender, about 3 more minutes.
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Add diced zucchini (reserve the sliced zucchini for the topping) and red pepper. Cook until the zucchini and pepper are tender, about 2 minutes. Season with 1 teaspoon of the salt and ½ teaspoon of the pepper.
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Add the spinach and stir to combine. Cook until the spinach is wilted and reduced. Remove from heat. Let the mixture cool for 5 minutes. Tilt the skillet and spoon off any liquid if necessary.
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Meanwhile, in a large bowl whisk the eggs and milk (or cream). Season with remaining salt and pepper and the nutmeg. Stir in the Rosemary, thyme and cheeses.
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Transfer the vegetable mixture to the prepared baking dish and spread it evenly. Pour the egg mixture over top. With a spatula gently stir the egg mixture to evenly combine with the vegetables. Layer the sliced zucchini in rows on top of the egg. Sprinkle top lightly with Parmesan cheese.
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Bake for 30 – 35 minutes until just lightly golden and set. Remove from oven and let cool for at least 10 – 15 minutes, or longer, before serving. Cut into squares.