Bring first 9 ingredients to a boil in a large Dutch oven; reduce heat, cover, and simmer 1 hour or until chicken is done. Remove chicken, reserving broth in Dutch oven. Let chicken cool. Pour broth through a fine wire mesh strainer into an 8-cup liquid measuring cup; discard solids.
Cook rice according to package directions, substituting 4 1/4 cups reserved chicken broth for water and omitting butter. (Add water to broth to equal 4 1/4 cups, if necessary.)
Skin, bone, and coarsely chop or shred chicken.
Melt 1/2 cup butter in a large Dutch oven over medium heat; add mushrooms and green onions, and sauté 10 minutes or until tender. Stir in rice, chicken, sour cream, and soup. Spoon mixture into 8 (2-cup) casserole dishes, 8 (5 1/2- x 3 1/2- x 2-inch) mini-loaf pans, 3 (8-inch or 9-inch) square baking dishes, or 1 (4-quart or 15- x 10-inch) casserole dish.
Stir together crushed crackers and fried onions. Stir in 1/4 cup melted butter, paprika, and garlic powder. Sprinkle casserole evenly with cracker mixture.
Bake, covered, at 350° for 20 to 30 minutes. Uncover and bake 5 to 10 more minutes or until bubbly.
To make ahead: Cover casseroles tightly with foil, and freeze unbaked up to 1 month. Thaw overnight in refrigerator. Bake as directed. (Casserole may also be baked frozen. Plan to double the baking times.)
To microwave one frozen, unbaked 8-inch square casserole: Casserole must be in a microwave-safe dish. Cover dish with wax paper. Microwave at HIGH 15 to 16 minutes or until casserole is bubbly, giving dish a half-turn once.
To microwave one thawed, unbaked 8-inch square casserole: Follow directions to microwave a frozen, unbaked casserole, reducing microwave time to 7 to 8 minutes.
Kitchen Express: Use 2 deli-roasted chickens (about 2 pounds each) in place of boiled chickens. Use 1 cup dry sherry and 3 1/4 cups (26 ounces) canned chicken broth to cook rice.