Put dried mushrooms in a small bowl with 1 cup very hot water, stir, and let stand until softened, about 10 minutes.
Meanwhile, cut onion into large chunks, then pulse in a food processor to chop. In a 5- to 6-qt. pan over medium-high heat, cook onion and button mushrooms with oil, pepper, and thyme, stirring occasionally, until lightly browned, 8 to 10 minutes. While onion mixture cooks, put broth in a microwave-safe bowl and microwave until steaming, about 3 minutes.
Add broth, rice, squash, beans, and 1 tsp. salt to pan. Cover and bring to a boil over high heat, then reduce heat and simmer until squash is tender when pierced, 8 to 10 minutes. Meanwhile, swish dried mushrooms in soaking liquid to loosen grit. Lift out mushrooms and coarsely chop. Add mushrooms to pot. Carefully pour in soaking liquid, leaving grit behind in bowl.
Rinse chard and chop leaves and stems. Add chard to stew and cook until stems are tender, 10 minutes. Add more salt if you like and serve with parmesan.
*Cooked wild rice is available from Trader Joe's; you can also cook your own (do it ahead and freeze in batches). Buy prepared squash in the produce section of many markets.