Place chicken in a dutch oven; add water and *Bouquet Garni (3 sprigs fresh parsley, 1 bay leaf, 6 whole black peppercorns, 1/2 teaspoon 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.
Cook broccoli according the package directions.
Arrange broccoli spears crosswise in a 13x9x2" baking dish.
Melt butter 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.
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 350 for 20 minutes or until casserole is thoroughly heated and top is golden brown.