1. Place chicken in a Dutch oven; add water and 3 sprigs fresh parsley, 1 bay leaf, 6 whole black peppercorns, 1/2 tsp dried whole thyme, all enclosed in a double layer of cheesecloth and tied securely. Cover and bring to a boil. Reduce heat, and simmer 25 minutes or until chicken is done. Remove chicken and Bouquet Garni from broth, reserving 2 cups broth. Discard Bouquet Garni. Chill reserved broth, and skim congealed fat from surface. Bone and slice chicken; set aside.
2. Cook broccoli according the package directions. Drain well. Arrange broccoli spears crosswise in a 13 inch x 9 inch x 2 inch baking dish. Set aside. Melt margerine in a large skillet over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add reserved broth; cook over medium heat, stirring constantly, until mixture is thickened and bubbly. Remove from heat; stir in milk, sherry, salt, and pepper.
3. Pour half of sauce over broccoli and top with chicken slices. Add 2 tablespoons cheese to remaining sauce in skillet; stir until well blended, and pour over chicken. Sprinkle with remaining 2 tablespoons cheese, and bake at 350F for 20 minutes or until casserole is thoroughlyheated and top is golden brown.