In a large pot, heat butter over medium heat. Add onions, mushrooms and carrots; sauté until onions are soft, about 10 minutes.
Meanwhile, in a medium saucepan, whisk milk and flour over medium heat, stirring frequently until milk starts to boil. Whisk constantly for one minute or until milk thickens. Pour into onion mixture and cook for 10 minutes, stirring occasionally. Stir in chicken broth and cooked rice; cook for 10 - 15 minutes (or until carrots are soft.)
Ladle into bowls and top each serving with 2 tablespoons cheese and 1 teaspoon parsley.