Cook asparagus in large pot of boiling salted water until crisp-tender, about 3 minutes. Using slotted spoon, transfer to bowl; set aside. Reserve water in pot.

Melt butter in large skillet over medium-low heat. Add prosciutto and stir 1 minute. Add mushrooms and sauté until golden, about 3 minutes. Add asparagus, peas, and cream and simmer until cream is reduced by 1/3, about 2 minutes. Turn off heat. Mix in cherry tomatoes.

Bring reserved water in pot to boil. Add pasta and cook until tender but still firm to bite, stirring occasionally. Drain. Add pasta to skillet with sauce and toss to coat over low heat. Season to taste with salt and pepper. Remove from heat. Stir in cheese. Transfer to large bowl; sprinkle with chives and serve.