Place the dried mushrooms in a glass bowl and cover with 1 cup hot water and let stand.
Clean and slice the fresh mushrooms and peel and chop the onion. Heat the olive oil in a large saucepan over medium heat. Add the sliced fresh mushrooms and cook until lightly browned, about 5 to 7 minutes.
While the fresh mushrooms are browning, strain the dried mushrooms through a paper-towel-lined sieve reserving the soaking liquid. Rinse and drain the mushrooms, then chop finely. Place the reserved mushroom liquid and the stock in a saucepan and heat to a simmer.
Add the butter to the browned mushrooms and let melt. Stir in the onion and the refreshed dried mushrooms and cook, stirring often until onion is soft but not browned, about 2-3 minutes. Stir in barley until well coated with the melted butter and oil from the pan. Add the fresh thyme and freshly ground pepper.
Add the white wine and bring to a boil, stirring until the wine has been absorbed. Stir in 2 cups (500 mL) of the stock and mushroom soaking liquid mixture. Increase heat to high and bring to a boil. Reduce heat to low and simmer, uncovered and stirring occasionally, for about 15 minutes or until stock has been absorbed, and barley has swollen and is starting to soften.
Add hot stock mixture to barley ½ cup (125 mL) at a time, stirring often and allowing barley to absorb stock before adding more, until most of stock is used (you may not need all of it) and barley is tender but still a little chewy, about 15 minutes.
Meanwhile, trim woody ends from asparagus by snapping spears where they break naturally. Cut asparagus diagonally into ½-inch (1-cm) pieces. Stir asparagus into barley. Cover and cook for 5 to 7 minutes or until asparagus is tender-crisp. Stir in the freshly grated parmesan cheese. Taste and adjust seasoning.