1. Cut prosciutto into 1/2 inch wide strips. Set aside. Cut green onions into 1 inch pieces. Set aside. Cut asparagus cross wise into 3/4 inch pieces. Set aside.
2. Melt butter with 2 tablespoons oil in heavy large skillet over medium heat. Add onions and shallot. Sprinkle with coarse salt and pepper. Sauté until tender (do not brown), about 8 minutes. Add wine; increase heat to medium-high and simmer until liquid is reduced to glaze, about 3 minutes. Add broth and bring to simmer; set aside.
3. Cook asparagus in large pot of boiling salted water until just tender, 2 to 4 minutes, depending on thickness of asparagus. Using skimmer or slotted spoon, transfer to large bowl of ice water. Return water to boil. Add peas and cook until just tender, about 2 minutes. Using skimmer, transfer to
bowl with asparagus. Drain vegetables.
4. Return water in pot to boil. Cook pasta until tender but still firm to bite, stirring occasionally. Drain, reserving 1 cup pasta cooking liquid.
5. Meanwhile, reheat onion mixture. Add lettuce and stir just until wilted, about 1 minute. Add drained asparagus and peas; stir until heated through.
6. Add pasta, 1 cup Parmesan cheese, and parsley to skillet with vegetables; toss, adding reserved pasta cooking liquid by 1/4 cupfuls if dry. Season with salt and pepper.
7. Transfer pasta to large shallow bowl. Sprinkle prosciutto over; drizzle with olive oil. Serve.