Butter 8x8x2-inch glass baking dish. Melt 1 tablespoon butter in large nonstick skillet over medium-high heat. Add mushrooms; saute until juices have cooked away, about 10 minutes. Transfer mushrooms to large bowl. Melt 1 tablespoon butter in same skillet over medium-high heat. Add onions; saute until brown and tender but still holding shape, about 6 minutes. Add to bowl with mushrooms. Melt 1 tablespoon butter in same skillet over medium heat. Add brussels sprouts; saute until browned, about 10 minutes. Add 1/2 cup broth to skillet. Cover and simmer until brussels sprouts are tender, stirring often, about 10 minutes. Uncover and cook until liquid evaporates, about 1 minute. Add to bowl with mushrooms. Melt 1 tablespoon butter in same skillet over medium heat. Add garlic and thyme; saute 1 minute. Add flour and stir 1 minute. Gradually add remaining 2 cups broth and cream and bring to boil, whisking often. Boil until reduced to 1 1/4 cups, about 8 minutes. Mix sauce into vegetables in bowl. Season with salt and pepper. Transfer vegetables to prepared dish. (Can be made 3 days ahead. Cover and chill.) Preheat oven to 375°F. Melt 1 tablespoon butter in small saucepan. Mix in breadcrumbs and cheese. Sprinkle crumb mixture over vegetables. Bake uncovered until vegetables are heated through and topping is golden, about 35 minutes.