Rinse mushrooms under cold running tap water to remove any debris. Place mushrooms in bowl, and cover with boiling water. Let soak 15 to 20 minutes. Rinse well, and chop.
Heat oil in skillet over medium-high heat. Add onion and garlic and cook, stirring occasionally, until soft, about 5 minutes. Add mushrooms and cook 5 minutes. Add tomatoes and sugar. Season to taste with salt and black pepper.
Bring to a boil and cook, stirring, 3 to 5 minutes. Add cream and one-third of basil. Taste, and adjust seasoning as necessary. Remove from heat.
Bring large pot of water to a boil. Add penne and cook until almost done, then add asparagus during final minute of cooking. Pasta should be al dente and asparagus just tender.
Toss hot pasta and asparagus with tomato mixture. Add Parmesan cheese and remaining basil; toss to combine. Serve immediately.