In a small saucepan, combine the dried shiitake mushrooms and 1 cup of the broth, bring to a boil, then remove from the heat and let sit for 10 minutes.

In a large pot over medium heat, warm the butter and oil. Add the onion and garlic and sauté until the onions are translucent and fragrant, 3 to 4 minutes. Add the mixed fresh mushrooms, thyme, and bay leaf. Cook until the mushrooms are soft and starting to release their liquid, about 6 minutes. Add the soaked shiitake mushrooms, strain and add their soaking liquid, then add the remaining 5 cups broth and the rice. Bring to a boil, then reduce the heat and simmer for 10 minutes. Season to taste with salt and pepper. Remove and discard the thyme sprigs and bay leaf. Stir in the vinegar and garnish with the parsley just before serving.