Melt the butter in a small skillet over medium heat. Add the onions, celery, and sausage and sauté until softened, about 5 minutes. Remove and set aside.
Cut the chicken into bite-sized pieces and pat it dry. Arrange on the bottom of a lightly greased 9 x 13-inch baking dish. Season with salt and pepper.
Combine the cream of chicken soup, sour cream, milk, thyme, and ground sage. Add it to the top of the chicken and spread evenly to coat.
Top with thawed green beans. Note: The green beans will be al dente (have a slight crunch). For softer green beans, submerge them into the sauce a bit.
Combine the stuffing mix, chicken broth, sauteed onions, celery, and sausage. Add it to the top of the casserole dish.
Cover and bake for 40 minutes. Remove the cover and bake for 5 more minutes.