Preheat the oven to 350 degrees F (175 degrees C).
Heat oil in a pan over medium-high heat. Cook and stir chicken in the hot oil until almost browned and no longer pink in the center, 5 to 7 minutes. Add mushrooms, onion, and parsley; saute until cooked down, about 5 minutes. Add paprika, celery salt, garlic powder, salt, and pepper.
Pour in 1/4 cup milk and turn down temperature to medium-low or low. Push food to the edges of the pan and put cream cheese in the middle to melt. Mix, adding more milk to desired consistency, increasing heat to thicken. Stir in broccoli and Parmesan cheese. Taste, adjust seasoning, and transfer to a casserole dish.
Melt butter in the same hot pan with residual heat. Pour in crushed corn flakes and mix until coated. Pour over casserole.
Bake in the preheated oven until bubbly, about 30 minutes.