Large portobello mushrooms take the place of meat in the hearty vegetarian dish.
Heat oven to 450 degrees. Spray two roasting pans with cooking spray. Roughly chop reserved stems of portobello and shiitake; transfer to one pan. Arrange mushroom caps in second pan. Roast both pans 15 to 20 minutes until juices have been released. Remove pan with mushroom caps from oven, and set aside. Remove pan with stems, add white wine, and cook, stirring, over high heat 2 to 3 minutes; set aside.
Place porcini mushrooms in saucepan. Pour stems and liquid from roasting pan into saucepan. Add 4 cups water and 4 sprigs thyme. Cover; simmer 45 minutes. Strain through fine sieve lined with dampened cheesecloth. Press down on solids with spatula to rid mushrooms of excess liquid. Set aside. Discard solids.
Cut portobello caps in half; reserve. Slice potatoes into four-inch-long strips one-inch wide; set aside in cold water.
Spray bottom of large skillet with cooking spray. Add shallots and garlic; cover, and saute over medium heat, turning, until brown, 5 to 8 minutes. Add reserved broth, remaining 4 sprigs thyme, salt, pepper, all reserved caps, and collected juices. Drain potatoes; add to broth. Simmer, covered, until potatoes and shallots are soft, 15 to 20 minutes, stirring once.
Meanwhile, blanch asparagus in boiling water just until bright green, 1 to 2 minutes. Transfer to ice water to stop cooking; drain. Add asparagus to broth; cook until heated through, 3 to 5 minutes. Garnish with parsley and shaved Parmesan.