Heat the oven: Preheat to 350°F (175°C). Lightly grease a 9x13-inch baking dish.
Prep the broccoli: If using fresh, steam or blanch for 2–3 minutes until bright green and crisp-tender.
If using frozen, thaw and pat dry. Excess moisture can make the casserole watery.
Make the “million dollar” sauce: In a large bowl, beat together cream cheese, sour cream, and cream of chicken soup until smooth. Stir in garlic, onion powder, paprika, thyme, salt, and pepper.
Add the cheese: Mix in 1 cup cheddar and all the Parmesan.
Reserve the remaining 1/2 cup cheddar for topping.
Combine: Fold in chicken and broccoli until evenly coated. Taste and adjust seasoning.
Assemble: Spread mixture into the baking dish. Sprinkle with remaining cheddar.
Make the topping: Toss crushed crackers (or panko) with melted butter.
Scatter evenly over the casserole.
Bake: 25–30 minutes, until hot and bubbling at the edges and the top is golden.
Rest and serve: Let it sit 5–10 minutes to set. Serve with rice, buttered noodles, or a crisp salad.