Preheat oven to 375 degrees. Grease a 2 quart baking dish with non-stick spray or butter and set aside.
Bring a large pot of salted water to a boil, add the broccoli, reduce heat to medium and simmer for 3 minutes. Drain and set aside.
While the broccoli is cooking, stir together the cream of chicken soup, mayonnaise, and seasonings, then whisk in the milk.
Pour the soup mixture over the cooked and drained broccoli along with the cheese. Stir well to combine and pour into prepared dish.
Bake for 15-20 minutes until bubbly.
Melt the butter in a medium bowl. Place the crackers in a large zip top bag and seal. Break the crackers up in the bag using your hands.
Stir the cracker crumbs into the butter until well combined.
Remove the casserole from the oven, top with the buttered crackers then return to the oven and bake for 5 more minutes.
Serve immediately. Store leftovers in the refrigerator for up to 3 days.