Preheat oven to 400 degrees. Butter a 10 inch baking dish.
Melt 2 Tablespoons butter in a large saute pan. Add the onion, saute until soft, add the chopped garlic to the pan, cook about one minute.
Add 2 Tablespoons butter to the pan with the onions and garlic. Cook until the butter melts, add the flour, stir to make a roux. Cook the roux, stirring while it cooks for 2 minutes until lightly browned. Add the chicken broth to the pan in a slow,steady stream while whisking. Add the milk to the pan slowly while whisking. Simmer until slightly thickened into a gravy consistency.
Mix the rotisserie chicken, gravy,broccoli, rice and half the cheese in a large bowl. Place the mixture into the buttered casserole dish. In a small bowl mix bread crumbs and melted butter, add the remaining cheese. Spread the mixture over top the casserole. Bake at 400 degrees for 20 minutes or until top is crisp and golden.