Preheat the oven to 375°F (190°C) and lightly grease a 9×13-inch baking dish.
In a large mixing bowl, whisk together the olive oil, chicken broth, heavy cream, paprika, garlic powder, and ground mustard until combined.
Add the frozen peas and carrots, frozen corn, and orzo to the bowl; stir to combine.
Stir in 1 cup shredded cheddar and 1/3 cup grated Parmesan; season with salt and pepper (about 1 teaspoon each, or to taste). Mix gently and press the mixture into an even layer in the prepared baking dish.
Sprinkle the top evenly with the remaining 1/2 cup cheddar and 1/3 cup Parmesan.
Bake, uncovered, for 40–45 minutes, until the orzo is tender and the casserole is creamy and set.
Remove the dish from the oven. Add the diced rotisserie chicken, finely diced broccoli, and the remaining 1/2 cup cheddar; gently fold them into the casserole.
Cover loosely with foil and let rest for 10 minutes to thicken, then stir gently once more and adjust seasoning if needed before serving.