1Heat the oven to 375°F and set a rack in the middle. Warm the broth in a small pot or in the microwave until steaming; hot liquid helps the raw rice start cooking right away.
2Pat the chicken dry and season it with salt, pepper and the paprika. Heat the olive oil in a large skillet over medium-high heat and brown the chicken in a single layer for about 3 minutes per side. It does not need to cook through. Move it to a plate.
3Lower the heat to medium and melt the butter in the same skillet. Add the onion, celery and carrots and cook for 5 minutes, then add the mushrooms and cook another 5 minutes until their liquid evaporates and they start to brown. Stir in the garlic and thyme for 30 seconds.
4Sprinkle the flour over the vegetables and stir for 1 minute. Slowly pour in the hot broth, then the milk, stirring and scraping up the browned bits, and simmer for 2 minutes until the sauce thickens slightly. Season with salt and pepper; it should taste a little salty, since the rice will absorb a lot.
5Spread the uncooked rice in an ungreased 9x13 baking dish. Scatter the browned chicken and any juices from the plate on top, then pour the hot sauce over everything. Gently press the chicken and rice down so nothing sticks up out of the liquid.
6Cover the dish tightly with foil and bake for 40 minutes. Uncover, stir once, and check that the rice is nearly tender and the chicken is cooked through. If the dish looks dry, splash in a few tablespoons of hot broth or water.
7Stir in the frozen peas, sprinkle the cheddar over the top, and bake uncovered for 10 to 12 minutes until the cheese is melted and bubbling at the edges and the rice is fully tender.
8Let the casserole rest for 10 minutes so the sauce sets, then scatter the parsley over the top and serve.