1. In a heatproof bowl, cover the dried porcini with the hot water and let the mushrooms soak until they are softened, about 20 minutes. Lift the porcini from the soaking liquid, then rinse and coarsely chop them. Slowly pour the porcini soaking liquid into a measuring cup, stopping just before you reach the grit at the bottom.
2. Melt 2 tablespoons of the unsalted butter in a medium skillet. Add the enoki mushrooms in an even layer, season with salt and pepper and cook over moderate heat, without stirring, until the mushrooms are golden brown, about 4 minutes. Stir the enoki and cook until evenly browned and crisp, about 1 minute longer. Using a slotted spoon, transfer the enoki mushrooms to a paper towel to drain.
3. Add 1/2 tablespoon of the olive oil to the medium skillet. Add the shiitake mushrooms and season with salt and pepper. Cover and cook the mushrooms over moderate heat until golden, about 4 minutes. Uncover and cook, stirring, until the mushrooms are browned all over, about 2 minutes longer. Cover and set aside.
4. In a medium saucepan, bring the chicken stock and the porcini soaking liquid to a boil. Cover the stock and reduce the heat to low.
5. In a large saucepan, heat the remaining 1 tablespoon of olive oil. Add the onion, garlic and thyme and cook over moderate heat until the onion is golden, about 4 minutes. Add the rice and cook, stirring, until coated with oil. Add the wine and simmer until almost evaporated, about 1 minute. Add enough hot stock to cover the rice and cook, stirring constantly, until it is absorbed. Continue to add the stock, 1 cup at a time, stirring constantly until it is absorbed between additions. The risotto is done when the rice is just tender and creamy, about 25 minutes total.
6. Stir in the reserved porcini and shiitake mushrooms, the remaining 1 tablespoon of butter and the 2 tablespoons of Parmesan cheese. Season with salt and pepper.
7. Add the enoki to the shiitake skillet. Cook over high heat, stirring, until hot. Spoon the mushroom risotto into shallow bowls. Top the risotto with the fried enoki mushrooms and chopped parsley and serve with more Parmesan cheese.
Make Ahead: The shiitake and enoki mushrooms can be prepared through Step 3 early in the day and kept at room temperature.