Preheat the oven to 350°F (175°C).
Steam the broccoli florets in a steamer basket over boiling water for 4-5 minutes until tender-crisp. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion, sliced mushrooms, and minced garlic. Sauté for 5-6 minutes until the vegetables are softened and fragrant. Remove from heat.
In a large mixing bowl, combine the cooked shredded chicken, steamed broccoli, sautéed vegetables, condensed cream of mushroom soup, sour cream, mayonnaise, salt, black pepper, and paprika. Mix well until fully incorporated.
Fold in the shredded cheddar cheese and mozzarella cheese, reserving 1/4 cup of cheddar for topping.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle the reserved cheddar cheese and grated Parmesan cheese evenly over the top.
Sprinkle the crushed Ritz crackers evenly over the cheese layer.
Bake uncovered for 35-40 minutes until the casserole is bubbly and the topping is golden brown.
Remove from oven and let it rest for 5 minutes before serving.