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Trim asparagus stems and cut stalks into 2-inch segments. If stalks are fairly thick, split in half lengthwise.
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Heat the butter and oil in a large skillet; add asparagus, onions, and mushrooms and sauté over medium-high heat for 2 minutes.
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Stir in garlic and white wine and cook 2-4 minutes longer, or until asparagus is bright green.
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Season with salt and pepper.
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As the asparagus cooks, place fresh fettuccine in boiling salted water until al dente, about 1 minute.
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Drain, reserving 1/2-cup cooking water. In a large bowl, combine pasta with asparagus mixture and cheese.
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Toss to mix, adding reserved pasta water if needed to keep moist.
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Stir in nasturtiums and serve immediately.