In a large heavy saucepan, cook mushrooms in the broth with onion and thyme until tender, about 10 to 15 minutes.
In blender or food processor, puree the mixture , leaving some chunks of vegetable in it. Set aside.
In the saucepan, melt the butter, whisk in the flour until smooth. Add the salt, pepper, rice milk and mushroom puree. Stirring constantly, bring soup to a boil. Add potato, carrots, celery and rice. Simmer 30 minutes or till potatoes and rice are tender. Adjust seasoning.