Preheat oven to 350 degrees F (175 degrees C). Butter a 9" x 13" baking dish.
In a medium mixing bowl, whisk together condensed cream of mushroom soup, mayonnaise, beaten egg, and onions.
Place frozen broccoli into a large mixing bowl and break up. Using a rubber spatula, scrape soup/mayonnaise mixture on top of broccoli and mix well. Sprinkle on shredded cheese and mix well. Spread mixture into prepared baking dish and smooth the top with spatula. Season top with salt, pepper and paprika.