Preheat oven to 350 degrees F (175 degrees C). Butter a 9x13 inch baking dish.
In a medium bowl, whisk together condensed cream of mushroom soup, mayonnaise, egg and onion.
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 to taste with salt, pepper and paprika.
Bake for 45 minutes to 1 hour in the preheated oven.