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