Preheat Oven and Prepare Baking Dish: Preheat oven to 425 degrees F (218 degrees C). Coat a 9x13-inch baking dish with nonstick cooking spray, then set aside.
Steam Broccoli: Arrange a steamer basket in a saucepan with about 1 inch of water in it. Put broccoli florets in steamer basket, cover with a lid, bring water to a boil over high heat, then steam broccoli for about 3-5 minutes until just tender, then set aside to cool slightly.
Mix Casserole Base: In a large mixing bowl, mix together cream of mushroom soup, eggs, sour cream or Greek yogurt, dried minced onion, salt, and pepper.
Add Broccoli and Cheese: Stir in steamed broccoli and half the shredded cheese.
Transfer to Baking Dish: Pour broccoli mixture into prepared baking dish, then use a rubber spatula to spread into an even layer.
Add Toppings: Sprinkle remaining shredded cheese on top. Then, in a small bowl, combine crushed crackers with melted butter and spread this over the cheese layer.
Bake: Transfer baking dish to oven and bake for 25-30 minutes until golden.
Serve: Allow to cool slightly before serving.