How to Make It
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.
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.
In a blender or food processor, puree the soup until completely smooth. Return the soup to the pot and stir in the reserved mushrooms.
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.
Reheat the soup if necessary. Serve topped with the Gruyère croûtes.