Using large non-stick sauté pan, heat olive oil over medium high heat. After 1 minute, add onion and garlic and sauté for about 2 minutes, or until onion has softened a bit. Add asparagus, squash, snow peas, mushrooms and broccoli, and continue to cook until vegetables begin to soften, about 3-4 minutes.
While vegetables are cooking, bring a large pot of salted water to a boil and cook pasta to al dente, or for about 7-8 minutes. Remove from the heat and drain (reserving a ½ cup of the pasta water).
Pour pasta into pan with vegetables and add baby peas, heavy cream, mozzarella, and parmesan. Beginning with ¼ cup, add reserved pasta water to pan and continue to mix well, add additional pasta water as necessary to bring sauce together. Season with salt and pepper and let pasta cook for a few more minutes. Serve and garnish with additional parmesan cheese at table.