Preheat oven to 350 degrees F (175 degrees C). Butter a 9×13 inch baking dish.
In a medium mixing bowl, whisk together condensed soup, mayonnaise, egg and onions.
Place frozen broccoli into a very large mixing bowl. (I like to use my large stainless steel bowl to mix this recipe thoroughly.) Break up the frozen broccoli. Using a rubber spatula, scrape soup-mayonnaise mixture on top of broccoli, and mix well. Sprinkle on cheese, and mix well. Spread mixture into prepared baking dish, and smooth top of casserole. Season with salt, pepper and paprika.
Bake for 45 minutes to 1 hour.