Preheat oven to 400°. In a heavy saucepan melt 3 1/2 tablespoons of butter over moderately low heat. Add flour and cook, whisking for 3 minutes. Add broth, bring mixture to a boil, and simmer it, whisking occasionally for 15 minutes.
While sauce is simmering, in a skillet cook shallots and mushrooms in the remaining 1 tablespoon of butter over moderate heat, stirring occasionally, until most of the liquid is evaporated.
Add mushroom mixture to sauce and stir in cream, nutmeg, lemon juice, chicken, cranberries or cherries, rice, and salt and pepper to taste.
Transfer mixture to a 13 x 9-inch baking dish and bake casserole covered, in the middle of preheated oven for 20 minutes.
Uncover, bake casserole for 15- 25 additional minutes, or until most of the liquid has been absorbed. Garnish casserole with additional cherries or cranberries.