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Preheat oven to 350F.
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Wash and dry chard. Remove stems and save for another use.Chop leaves to a small size.
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Chop onion. Heat the olive oil in a large frying pan . Add onion and saute 4-5 minutes on medium-high heat, just until the onions are barely starting to brown. Then add Swiss chard, all at once. Turn the chard as it cooks until it's all wilted, about 2-3 minutes. Turn off heat and let the chard cool slightly.
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Combine the milk and goat cheese.
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In a large bowl, beat the eggs, then season with salt and fresh ground black pepper. Stir the milk/goat cheese mixture into the eggs and then add the mozzarella and two tablespoons of the Parmesan.
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Add the chard and onions and stir so it's well distributed in the egg mixture.
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Spray a round casserole dish or spring form pan with olive oil or nonstick spray. (Use a pan that's 8 or 9 inches in diameter.)
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Put the egg/chard mixture into the pan and top with the other two tablespoons of parmesan.
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Bake for one hour, or until the mixture is fully set and lightly browned on top.