Preheat the oven to 375°F (190°C) and lightly grease a 9×13-inch baking dish.
In the prepared dish, whisk together the olive oil, chicken broth, and heavy cream until combined.
Add the paprika, garlic powder, and ground mustard to the liquid and stir to distribute the seasonings.
Stir in the frozen peas and carrots, frozen corn, and dry orzo until evenly mixed with the liquid.
Fold in 1 cup shredded Cheddar and 1/3 cup grated Parmesan, then season with salt and pepper (about 1 teaspoon each suggested), and spread the mixture into an even layer in the dish.
Sprinkle the top 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 bubbling around the edges.
Remove from the oven and add the diced rotisserie chicken, finely diced broccoli, and the remaining 1/2 cup Cheddar; gently stir to combine.
Cover the dish with foil and let rest for 10 minutes to thicken, then stir gently once more and adjust seasoning to taste before serving warm.