Step 1 Heat the oven to 350 degrees.
Step 2 In a large saute pan heated over medium-high heat, melt the butter. Stir in the onions and mushrooms and cook, stirring frequently, until the onions are translucent, 6 to 8 minutes. Remove from heat.
Step 3In a large bowl, stir together the mushroom soup, mayonnaise, eggs, 1 cup cheddar and 1 cup Parmesan cheese until thoroughly combined. Stir in the thawed broccoli, then the cooked onions and mushrooms. Place the mixture into a 13-by-9-inch baking dish.
Step 4In a medium bowl, combine the remaining cheddar and parmesan cheese along with the French-fried onions to form the topping. Scatter the topping evenly over the broccoli.
Step 5Bake the casserole until bubbly and the topping is golden-brown, about 45 minutes. Cool slightly before serving.