Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium-high heat. Add the chicken pieces and cook until browned and cooked through, about 6-8 minutes. Remove chicken from skillet and set aside.
In the same skillet, add diced onion and sauté for 3-4 minutes until translucent. Add minced garlic and cook for 1 minute until fragrant.
Add broccoli florets and zucchini slices to the skillet. Cook for 5 minutes, stirring occasionally, until vegetables are slightly tender. Remove from heat.
In a medium saucepan, melt butter over medium heat. Whisk in flour and cook for 1-2 minutes to form a roux.
Gradually whisk in milk and heavy cream, stirring constantly until the sauce thickens, about 5 minutes.
Remove sauce from heat and stir in shredded cheddar cheese and half of the mozzarella cheese until melted and smooth. Season with salt, black pepper, and paprika.
In a large mixing bowl, combine cooked chicken, sautéed vegetables, and cheese sauce. Mix well to evenly coat all ingredients.
Transfer the mixture to a greased 9×13 inch baking dish.
In a small bowl, combine breadcrumbs, grated Parmesan cheese, and remaining mozzarella cheese. Sprinkle evenly over the casserole.
Bake in the preheated oven for 25-30 minutes, until the topping is golden brown and the casserole is bubbly.
Remove from oven and let cool for 5 minutes before serving.