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In a 2-3 quart pot, warm the butter and oil over medium heat; saute the garlic and onion until the onion is wilted, about 5 minutes.
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Add the mushrooms and saute, still over medium heat, until their liquid has exuded, then evaporated.
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Add the broth, water, and tomato paste; bring to a boil, adjust heat, and simmer gently, uncovered, for 5 minutes.
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In a small bowl, whisk the yolks until they are thick and lemony in color.
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Add the grated Parmesan cheese and parsley; beat well again.
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With the soup barely simmering, add the yolk mixture gradually, whisking constantly and vigorously over medium heat until the yolks and soup bind together in a light cream, about 3 minutes.
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Taste and season to taste with salt and pepper.
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Serve immediately in deep or flat bowls.