Preheat oven to 400 degrees F. Spray a 2-quart baking dish with cooking spray and set aside.
Steam the broccoli florets in the microwave just until crisp-tender (about 3 minutes); drain and set aside.
In a large skillet, melt butter over medium-high heat. Add apple, onion and celery; sauté just until tender (about 8-10 minutes).
Meanwhile, prepare stuffing according to package instructions. Use a fork to gently combine stuffing with apple, onion and celery mixture. Set aside.
In the prepared dish, stir together chicken, condensed soup, milk, sour cream and broccoli. Season with salt and pepper, to taste. If the sauce seems too thick, you can add another splash or so of milk until it reaches your desired consistency. Top with stuffing mixture.
Bake (uncovered) for 20-25 minutes, or until casserole is heated through.