1. In a large pot, heat 2 tablespoons of the oil over moderately high heat. Add the mushrooms, 1/4 teaspoon of the salt, and the pepper and cook, stirring occasionally, until the mushrooms are golden, about 5 minutes. Remove the mushrooms from the pot and set aside. Reduce the heat to moderately low and add another 1 1/2 tablespoons of the oil to the pot. Add the onion and cook, stirring occasionally, until translucent, about 5 minutes.
2. Add the water, broth, rice, and the remaining 2 teaspoons of salt to the pot. Bring to a boil. Continue boiling for 10 minutes, stirring occasionally. Add the asparagus. Cook until the asparagus is tender, about 5 minutes.
3. In a blender or food processor, puree the soup until completely smooth. Return the soup to the pot and stir in the reserved mushrooms.
4. Meanwhile, heat the broiler. Put the bread on a baking sheet and brush both sides of the bread with the remaining 3 tablespoons of oil. Broil the bread until brown, about 2 minutes. Turn and top with the Gruyère. Broil until the cheese melts, about 2 minutes longer.
5. Reheat the soup if necessary. Serve topped with the Gruyère croûtes.
Wine Recommendation: The wines of southern Italy, long ignored by wine lovers, are now recognized as among the most interesting the country has to offer. A full-bodied wine such as a white Greco di Tufo, an earthy, honeyed delight, will be terrific here.